Polytope of Type {3,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4}*1152c
if this polytope has a name.
Group : SmallGroup(1152,157852)
Rank : 4
Schlafli Type : {3,6,4}
Number of vertices, edges, etc : 3, 72, 96, 32
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,4}*576b
16-fold quotients : {3,6,2}*72
48-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s2*s3*s2*s3> of order 2.
16 facets:
16 of {3,6}*36
3 vertex figures:
2 of 2-fold non-regular quotient of {6,4}*384a
1 of 2-fold non-regular quotient of {6,4}*384a
P/N, where N=<s0*s2*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1*s2> of order 2.
16 facets:
16 of {3,6}*36
3 vertex figures:
3 of 2-fold non-regular quotient of {6,4}*384a
P/N, where N=<s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1*s2*s3> of order 2.
16 facets:
16 of {3,6}*36
3 vertex figures:
3 of 2-fold non-regular quotient of {6,4}*384a
P/N, where N=<s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1*s3> of order 2.
16 facets:
16 of {3,6}*36
3 vertex figures:
2 of 2-fold non-regular quotient of {6,4}*384a
1 of 2-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s1*s2*s3*s2*s1*s2*s3, s0*s2*s1*s2*s3*s2*s1*s0*s2*s3> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
3 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s1*s2*s1*s2*s3*s2*s1*s3*s2*s1> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of {6,4}*96
P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2, s0*s2*s1*s2*s3*s2*s1*s0*s3*s2> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s1*s0*s2*s1*s2*s3*s2*s1*s0*s3*s2*s1> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
3 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s1*s2*s1*s2*s3*s2*s1*s3*s2*s1*s3> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s1*s2*s3*s2*s1*s3*s2*s3, s0*s2*s1*s2*s3*s2*s1*s0*s3*s2*s3> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s1*s0*s2*s1*s2*s3*s2*s1*s0*s3*s2*s1*s3> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1*s2*s3> of order 4.
8 facets:
8 of {3,6}*36
3 vertex figures:
2 of 4-fold non-regular quotient of {6,4}*384a
1 of 4-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s0*s1*s2*s1*s0*s3*s2*s1*s3*s2*s1*s3, s1*s0*s2*s1*s2*s3*s2*s1*s0*s3*s2*s1> of order 8.
4 facets:
4 of {3,6}*36
3 vertex figures:
3 of 8-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s1*s2*s1*s2*s3*s2*s1*s3*s2*s1*s3, s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1> of order 8.
4 facets:
4 of {3,6}*36
3 vertex figures:
2 of 8-fold non-regular quotient of {6,4}*384a
1 of 8-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s2*s1*s2*s3*s2*s1*s2*s3, s0*s2*s1*s2*s3*s2*s1*s0*s2*s3> of order 8.
4 facets:
4 of {3,6}*36
3 vertex figures:
3 of 8-fold non-regular quotient of {6,4}*384a
P/N, where N=<s2*s3*s2*s3, s1*s2*s1*s2*s3*s2*s1*s3*s2*s1, s0*s1*s0*s2*s3*s2*s1*s0*s2*s3*s2*s1> of order 8.
4 facets:
4 of {3,6}*36
3 vertex figures:
2 of 8-fold non-regular quotient of {6,4}*384a
1 of 2-fold non-regular quotient of {6,4}*96
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,109)( 58,110)( 59,112)( 60,111)( 61,105)( 62,106)( 63,108)( 64,107)( 65,129)( 66,130)( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,141)( 74,142)( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,113)( 82,114)( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,125)( 90,126)( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(193,241)(194,242)(195,244)(196,243)(197,245)(198,246)(199,248)(200,247)(201,253)(202,254)(203,256)(204,255)(205,249)(206,250)(207,252)(208,251)(209,273)(210,274)(211,276)(212,275)(213,277)(214,278)(215,280)(216,279)(217,285)(218,286)(219,288)(220,287)(221,281)(222,282)(223,284)(224,283)(225,257)(226,258)(227,260)(228,259)(229,261)(230,262)(231,264)(232,263)(233,269)(234,270)(235,272)(236,271)(237,265)(238,266)(239,268)(240,267);;
s1 := ( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 61)( 6, 62)( 7, 64)( 8, 63)( 9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)( 16, 55)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 93)( 22, 94)( 23, 96)( 24, 95)( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 85)( 30, 86)( 31, 88)( 32, 87)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)( 47, 72)( 48, 71)( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(113,129)(114,130)(115,132)(116,131)(117,141)(118,142)(119,144)(120,143)(121,137)(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)(128,135)(145,193)(146,194)(147,196)(148,195)(149,205)(150,206)(151,208)(152,207)(153,201)(154,202)(155,204)(156,203)(157,197)(158,198)(159,200)(160,199)(161,225)(162,226)(163,228)(164,227)(165,237)(166,238)(167,240)(168,239)(169,233)(170,234)(171,236)(172,235)(173,229)(174,230)(175,232)(176,231)(177,209)(178,210)(179,212)(180,211)(181,221)(182,222)(183,224)(184,223)(185,217)(186,218)(187,220)(188,219)(189,213)(190,214)(191,216)(192,215)(243,244)(245,253)(246,254)(247,256)(248,255)(251,252)(257,273)(258,274)(259,276)(260,275)(261,285)(262,286)(263,288)(264,287)(265,281)(266,282)(267,284)(268,283)(269,277)(270,278)(271,280)(272,279);;
s2 := ( 2, 5)( 3, 9)( 4, 13)( 7, 10)( 8, 14)( 12, 15)( 17, 33)( 18, 37)( 19, 41)( 20, 45)( 21, 34)( 22, 38)( 23, 42)( 24, 46)( 25, 35)( 26, 39)( 27, 43)( 28, 47)( 29, 36)( 30, 40)( 31, 44)( 32, 48)( 49, 65)( 50, 69)( 51, 73)( 52, 77)( 53, 66)( 54, 70)( 55, 74)( 56, 78)( 57, 67)( 58, 71)( 59, 75)( 60, 79)( 61, 68)( 62, 72)( 63, 76)( 64, 80)( 82, 85)( 83, 89)( 84, 93)( 87, 90)( 88, 94)( 92, 95)( 97,129)( 98,133)( 99,137)(100,141)(101,130)(102,134)(103,138)(104,142)(105,131)(106,135)(107,139)(108,143)(109,132)(110,136)(111,140)(112,144)(114,117)(115,121)(116,125)(119,122)(120,126)(124,127)(146,149)(147,153)(148,157)(151,154)(152,158)(156,159)(161,177)(162,181)(163,185)(164,189)(165,178)(166,182)(167,186)(168,190)(169,179)(170,183)(171,187)(172,191)(173,180)(174,184)(175,188)(176,192)(193,209)(194,213)(195,217)(196,221)(197,210)(198,214)(199,218)(200,222)(201,211)(202,215)(203,219)(204,223)(205,212)(206,216)(207,220)(208,224)(226,229)(227,233)(228,237)(231,234)(232,238)(236,239)(241,273)(242,277)(243,281)(244,285)(245,274)(246,278)(247,282)(248,286)(249,275)(250,279)(251,283)(252,287)(253,276)(254,280)(255,284)(256,288)(258,261)(259,265)(260,269)(263,266)(264,270)(268,271);;
s3 := ( 1,146)( 2,145)( 3,148)( 4,147)( 5,150)( 6,149)( 7,152)( 8,151)( 9,154)( 10,153)( 11,156)( 12,155)( 13,158)( 14,157)( 15,160)( 16,159)( 17,162)( 18,161)( 19,164)( 20,163)( 21,166)( 22,165)( 23,168)( 24,167)( 25,170)( 26,169)( 27,172)( 28,171)( 29,174)( 30,173)( 31,176)( 32,175)( 33,178)( 34,177)( 35,180)( 36,179)( 37,182)( 38,181)( 39,184)( 40,183)( 41,186)( 42,185)( 43,188)( 44,187)( 45,190)( 46,189)( 47,192)( 48,191)( 49,194)( 50,193)( 51,196)( 52,195)( 53,198)( 54,197)( 55,200)( 56,199)( 57,202)( 58,201)( 59,204)( 60,203)( 61,206)( 62,205)( 63,208)( 64,207)( 65,210)( 66,209)( 67,212)( 68,211)( 69,214)( 70,213)( 71,216)( 72,215)( 73,218)( 74,217)( 75,220)( 76,219)( 77,222)( 78,221)( 79,224)( 80,223)( 81,226)( 82,225)( 83,228)( 84,227)( 85,230)( 86,229)( 87,232)( 88,231)( 89,234)( 90,233)( 91,236)( 92,235)( 93,238)( 94,237)( 95,240)( 96,239)( 97,242)( 98,241)( 99,244)(100,243)(101,246)(102,245)(103,248)(104,247)(105,250)(106,249)(107,252)(108,251)(109,254)(110,253)(111,256)(112,255)(113,258)(114,257)(115,260)(116,259)(117,262)(118,261)(119,264)(120,263)(121,266)(122,265)(123,268)(124,267)(125,270)(126,269)(127,272)(128,271)(129,274)(130,273)(131,276)(132,275)(133,278)(134,277)(135,280)(136,279)(137,282)(138,281)(139,284)(140,283)(141,286)(142,285)(143,288)(144,287);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,109)( 58,110)( 59,112)( 60,111)( 61,105)( 62,106)( 63,108)( 64,107)( 65,129)( 66,130)( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,141)( 74,142)( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,113)( 82,114)( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,125)( 90,126)( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(193,241)(194,242)(195,244)(196,243)(197,245)(198,246)(199,248)(200,247)(201,253)(202,254)(203,256)(204,255)(205,249)(206,250)(207,252)(208,251)(209,273)(210,274)(211,276)(212,275)(213,277)(214,278)(215,280)(216,279)(217,285)(218,286)(219,288)(220,287)(221,281)(222,282)(223,284)(224,283)(225,257)(226,258)(227,260)(228,259)(229,261)(230,262)(231,264)(232,263)(233,269)(234,270)(235,272)(236,271)(237,265)(238,266)(239,268)(240,267);
s1 := Sym(288)!( 1, 49)( 2, 50)( 3, 52)( 4, 51)( 5, 61)( 6, 62)( 7, 64)( 8, 63)( 9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)( 16, 55)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 93)( 22, 94)( 23, 96)( 24, 95)( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 85)( 30, 86)( 31, 88)( 32, 87)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)( 47, 72)( 48, 71)( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(113,129)(114,130)(115,132)(116,131)(117,141)(118,142)(119,144)(120,143)(121,137)(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)(128,135)(145,193)(146,194)(147,196)(148,195)(149,205)(150,206)(151,208)(152,207)(153,201)(154,202)(155,204)(156,203)(157,197)(158,198)(159,200)(160,199)(161,225)(162,226)(163,228)(164,227)(165,237)(166,238)(167,240)(168,239)(169,233)(170,234)(171,236)(172,235)(173,229)(174,230)(175,232)(176,231)(177,209)(178,210)(179,212)(180,211)(181,221)(182,222)(183,224)(184,223)(185,217)(186,218)(187,220)(188,219)(189,213)(190,214)(191,216)(192,215)(243,244)(245,253)(246,254)(247,256)(248,255)(251,252)(257,273)(258,274)(259,276)(260,275)(261,285)(262,286)(263,288)(264,287)(265,281)(266,282)(267,284)(268,283)(269,277)(270,278)(271,280)(272,279);
s2 := Sym(288)!( 2, 5)( 3, 9)( 4, 13)( 7, 10)( 8, 14)( 12, 15)( 17, 33)( 18, 37)( 19, 41)( 20, 45)( 21, 34)( 22, 38)( 23, 42)( 24, 46)( 25, 35)( 26, 39)( 27, 43)( 28, 47)( 29, 36)( 30, 40)( 31, 44)( 32, 48)( 49, 65)( 50, 69)( 51, 73)( 52, 77)( 53, 66)( 54, 70)( 55, 74)( 56, 78)( 57, 67)( 58, 71)( 59, 75)( 60, 79)( 61, 68)( 62, 72)( 63, 76)( 64, 80)( 82, 85)( 83, 89)( 84, 93)( 87, 90)( 88, 94)( 92, 95)( 97,129)( 98,133)( 99,137)(100,141)(101,130)(102,134)(103,138)(104,142)(105,131)(106,135)(107,139)(108,143)(109,132)(110,136)(111,140)(112,144)(114,117)(115,121)(116,125)(119,122)(120,126)(124,127)(146,149)(147,153)(148,157)(151,154)(152,158)(156,159)(161,177)(162,181)(163,185)(164,189)(165,178)(166,182)(167,186)(168,190)(169,179)(170,183)(171,187)(172,191)(173,180)(174,184)(175,188)(176,192)(193,209)(194,213)(195,217)(196,221)(197,210)(198,214)(199,218)(200,222)(201,211)(202,215)(203,219)(204,223)(205,212)(206,216)(207,220)(208,224)(226,229)(227,233)(228,237)(231,234)(232,238)(236,239)(241,273)(242,277)(243,281)(244,285)(245,274)(246,278)(247,282)(248,286)(249,275)(250,279)(251,283)(252,287)(253,276)(254,280)(255,284)(256,288)(258,261)(259,265)(260,269)(263,266)(264,270)(268,271);
s3 := Sym(288)!( 1,146)( 2,145)( 3,148)( 4,147)( 5,150)( 6,149)( 7,152)( 8,151)( 9,154)( 10,153)( 11,156)( 12,155)( 13,158)( 14,157)( 15,160)( 16,159)( 17,162)( 18,161)( 19,164)( 20,163)( 21,166)( 22,165)( 23,168)( 24,167)( 25,170)( 26,169)( 27,172)( 28,171)( 29,174)( 30,173)( 31,176)( 32,175)( 33,178)( 34,177)( 35,180)( 36,179)( 37,182)( 38,181)( 39,184)( 40,183)( 41,186)( 42,185)( 43,188)( 44,187)( 45,190)( 46,189)( 47,192)( 48,191)( 49,194)( 50,193)( 51,196)( 52,195)( 53,198)( 54,197)( 55,200)( 56,199)( 57,202)( 58,201)( 59,204)( 60,203)( 61,206)( 62,205)( 63,208)( 64,207)( 65,210)( 66,209)( 67,212)( 68,211)( 69,214)( 70,213)( 71,216)( 72,215)( 73,218)( 74,217)( 75,220)( 76,219)( 77,222)( 78,221)( 79,224)( 80,223)( 81,226)( 82,225)( 83,228)( 84,227)( 85,230)( 86,229)( 87,232)( 88,231)( 89,234)( 90,233)( 91,236)( 92,235)( 93,238)( 94,237)( 95,240)( 96,239)( 97,242)( 98,241)( 99,244)(100,243)(101,246)(102,245)(103,248)(104,247)(105,250)(106,249)(107,252)(108,251)(109,254)(110,253)(111,256)(112,255)(113,258)(114,257)(115,260)(116,259)(117,262)(118,261)(119,264)(120,263)(121,266)(122,265)(123,268)(124,267)(125,270)(126,269)(127,272)(128,271)(129,274)(130,273)(131,276)(132,275)(133,278)(134,277)(135,280)(136,279)(137,282)(138,281)(139,284)(140,283)(141,286)(142,285)(143,288)(144,287);
poly := sub<Sym(288)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
References : None.
to this polytope