Polytope of Type {6,8,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,6}*1152b
if this polytope has a name.
Group : SmallGroup(1152,157603)
Rank : 4
Schlafli Type : {6,8,6}
Number of vertices, edges, etc : 12, 48, 48, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
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 : {3,8,6}*576, {6,4,6}*576b
3-fold quotients : {6,8,2}*384b
4-fold quotients : {3,4,6}*288
6-fold quotients : {3,8,2}*192, {6,4,2}*192
8-fold quotients : {6,2,6}*144
12-fold quotients : {3,4,2}*96, {6,4,2}*96b, {6,4,2}*96c
16-fold quotients : {3,2,6}*72, {6,2,3}*72
24-fold quotients : {3,4,2}*48, {2,2,6}*48, {6,2,2}*48
32-fold quotients : {3,2,3}*36
48-fold quotients : {2,2,3}*24, {3,2,2}*24
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)( 30, 56)( 31, 53)
( 32, 54)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)( 39, 61)
( 40, 62)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 71)( 46, 72)( 47, 69)
( 48, 70)( 73, 74)( 77, 80)( 78, 79)( 81, 82)( 85, 88)( 86, 87)( 89, 90)
( 93, 96)( 94, 95)( 97,122)( 98,121)( 99,123)(100,124)(101,128)(102,127)
(103,126)(104,125)(105,130)(106,129)(107,131)(108,132)(109,136)(110,135)
(111,134)(112,133)(113,138)(114,137)(115,139)(116,140)(117,144)(118,143)
(119,142)(120,141)(147,148)(149,151)(150,152)(155,156)(157,159)(158,160)
(163,164)(165,167)(166,168)(169,193)(170,194)(171,196)(172,195)(173,199)
(174,200)(175,197)(176,198)(177,201)(178,202)(179,204)(180,203)(181,207)
(182,208)(183,205)(184,206)(185,209)(186,210)(187,212)(188,211)(189,215)
(190,216)(191,213)(192,214)(217,218)(221,224)(222,223)(225,226)(229,232)
(230,231)(233,234)(237,240)(238,239)(241,266)(242,265)(243,267)(244,268)
(245,272)(246,271)(247,270)(248,269)(249,274)(250,273)(251,275)(252,276)
(253,280)(254,279)(255,278)(256,277)(257,282)(258,281)(259,283)(260,284)
(261,288)(262,287)(263,286)(264,285);;
s1 := ( 1,169)( 2,170)( 3,175)( 4,176)( 5,174)( 6,173)( 7,171)( 8,172)
( 9,177)( 10,178)( 11,183)( 12,184)( 13,182)( 14,181)( 15,179)( 16,180)
( 17,185)( 18,186)( 19,191)( 20,192)( 21,190)( 22,189)( 23,187)( 24,188)
( 25,145)( 26,146)( 27,151)( 28,152)( 29,150)( 30,149)( 31,147)( 32,148)
( 33,153)( 34,154)( 35,159)( 36,160)( 37,158)( 38,157)( 39,155)( 40,156)
( 41,161)( 42,162)( 43,167)( 44,168)( 45,166)( 46,165)( 47,163)( 48,164)
( 49,193)( 50,194)( 51,199)( 52,200)( 53,198)( 54,197)( 55,195)( 56,196)
( 57,201)( 58,202)( 59,207)( 60,208)( 61,206)( 62,205)( 63,203)( 64,204)
( 65,209)( 66,210)( 67,215)( 68,216)( 69,214)( 70,213)( 71,211)( 72,212)
( 73,242)( 74,241)( 75,248)( 76,247)( 77,245)( 78,246)( 79,244)( 80,243)
( 81,250)( 82,249)( 83,256)( 84,255)( 85,253)( 86,254)( 87,252)( 88,251)
( 89,258)( 90,257)( 91,264)( 92,263)( 93,261)( 94,262)( 95,260)( 96,259)
( 97,218)( 98,217)( 99,224)(100,223)(101,221)(102,222)(103,220)(104,219)
(105,226)(106,225)(107,232)(108,231)(109,229)(110,230)(111,228)(112,227)
(113,234)(114,233)(115,240)(116,239)(117,237)(118,238)(119,236)(120,235)
(121,266)(122,265)(123,272)(124,271)(125,269)(126,270)(127,268)(128,267)
(129,274)(130,273)(131,280)(132,279)(133,277)(134,278)(135,276)(136,275)
(137,282)(138,281)(139,288)(140,287)(141,285)(142,286)(143,284)(144,283);;
s2 := ( 1, 75)( 2, 76)( 3, 74)( 4, 73)( 5, 79)( 6, 80)( 7, 78)( 8, 77)
( 9, 91)( 10, 92)( 11, 90)( 12, 89)( 13, 95)( 14, 96)( 15, 94)( 16, 93)
( 17, 83)( 18, 84)( 19, 82)( 20, 81)( 21, 87)( 22, 88)( 23, 86)( 24, 85)
( 25, 99)( 26,100)( 27, 98)( 28, 97)( 29,103)( 30,104)( 31,102)( 32,101)
( 33,115)( 34,116)( 35,114)( 36,113)( 37,119)( 38,120)( 39,118)( 40,117)
( 41,107)( 42,108)( 43,106)( 44,105)( 45,111)( 46,112)( 47,110)( 48,109)
( 49,123)( 50,124)( 51,122)( 52,121)( 53,127)( 54,128)( 55,126)( 56,125)
( 57,139)( 58,140)( 59,138)( 60,137)( 61,143)( 62,144)( 63,142)( 64,141)
( 65,131)( 66,132)( 67,130)( 68,129)( 69,135)( 70,136)( 71,134)( 72,133)
(145,219)(146,220)(147,218)(148,217)(149,223)(150,224)(151,222)(152,221)
(153,235)(154,236)(155,234)(156,233)(157,239)(158,240)(159,238)(160,237)
(161,227)(162,228)(163,226)(164,225)(165,231)(166,232)(167,230)(168,229)
(169,243)(170,244)(171,242)(172,241)(173,247)(174,248)(175,246)(176,245)
(177,259)(178,260)(179,258)(180,257)(181,263)(182,264)(183,262)(184,261)
(185,251)(186,252)(187,250)(188,249)(189,255)(190,256)(191,254)(192,253)
(193,267)(194,268)(195,266)(196,265)(197,271)(198,272)(199,270)(200,269)
(201,283)(202,284)(203,282)(204,281)(205,287)(206,288)(207,286)(208,285)
(209,275)(210,276)(211,274)(212,273)(213,279)(214,280)(215,278)(216,277);;
s3 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 25, 33)( 26, 34)( 27, 35)( 28, 36)( 29, 37)( 30, 38)( 31, 39)( 32, 40)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 73, 81)( 74, 82)( 75, 83)( 76, 84)( 77, 85)( 78, 86)( 79, 87)( 80, 88)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)
(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)
(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)(176,184)
(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)
(217,225)(218,226)(219,227)(220,228)(221,229)(222,230)(223,231)(224,232)
(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)(248,256)
(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)(272,280);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 3, 4)( 5, 7)( 6, 8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)( 30, 56)
( 31, 53)( 32, 54)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)
( 39, 61)( 40, 62)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 71)( 46, 72)
( 47, 69)( 48, 70)( 73, 74)( 77, 80)( 78, 79)( 81, 82)( 85, 88)( 86, 87)
( 89, 90)( 93, 96)( 94, 95)( 97,122)( 98,121)( 99,123)(100,124)(101,128)
(102,127)(103,126)(104,125)(105,130)(106,129)(107,131)(108,132)(109,136)
(110,135)(111,134)(112,133)(113,138)(114,137)(115,139)(116,140)(117,144)
(118,143)(119,142)(120,141)(147,148)(149,151)(150,152)(155,156)(157,159)
(158,160)(163,164)(165,167)(166,168)(169,193)(170,194)(171,196)(172,195)
(173,199)(174,200)(175,197)(176,198)(177,201)(178,202)(179,204)(180,203)
(181,207)(182,208)(183,205)(184,206)(185,209)(186,210)(187,212)(188,211)
(189,215)(190,216)(191,213)(192,214)(217,218)(221,224)(222,223)(225,226)
(229,232)(230,231)(233,234)(237,240)(238,239)(241,266)(242,265)(243,267)
(244,268)(245,272)(246,271)(247,270)(248,269)(249,274)(250,273)(251,275)
(252,276)(253,280)(254,279)(255,278)(256,277)(257,282)(258,281)(259,283)
(260,284)(261,288)(262,287)(263,286)(264,285);
s1 := Sym(288)!( 1,169)( 2,170)( 3,175)( 4,176)( 5,174)( 6,173)( 7,171)
( 8,172)( 9,177)( 10,178)( 11,183)( 12,184)( 13,182)( 14,181)( 15,179)
( 16,180)( 17,185)( 18,186)( 19,191)( 20,192)( 21,190)( 22,189)( 23,187)
( 24,188)( 25,145)( 26,146)( 27,151)( 28,152)( 29,150)( 30,149)( 31,147)
( 32,148)( 33,153)( 34,154)( 35,159)( 36,160)( 37,158)( 38,157)( 39,155)
( 40,156)( 41,161)( 42,162)( 43,167)( 44,168)( 45,166)( 46,165)( 47,163)
( 48,164)( 49,193)( 50,194)( 51,199)( 52,200)( 53,198)( 54,197)( 55,195)
( 56,196)( 57,201)( 58,202)( 59,207)( 60,208)( 61,206)( 62,205)( 63,203)
( 64,204)( 65,209)( 66,210)( 67,215)( 68,216)( 69,214)( 70,213)( 71,211)
( 72,212)( 73,242)( 74,241)( 75,248)( 76,247)( 77,245)( 78,246)( 79,244)
( 80,243)( 81,250)( 82,249)( 83,256)( 84,255)( 85,253)( 86,254)( 87,252)
( 88,251)( 89,258)( 90,257)( 91,264)( 92,263)( 93,261)( 94,262)( 95,260)
( 96,259)( 97,218)( 98,217)( 99,224)(100,223)(101,221)(102,222)(103,220)
(104,219)(105,226)(106,225)(107,232)(108,231)(109,229)(110,230)(111,228)
(112,227)(113,234)(114,233)(115,240)(116,239)(117,237)(118,238)(119,236)
(120,235)(121,266)(122,265)(123,272)(124,271)(125,269)(126,270)(127,268)
(128,267)(129,274)(130,273)(131,280)(132,279)(133,277)(134,278)(135,276)
(136,275)(137,282)(138,281)(139,288)(140,287)(141,285)(142,286)(143,284)
(144,283);
s2 := Sym(288)!( 1, 75)( 2, 76)( 3, 74)( 4, 73)( 5, 79)( 6, 80)( 7, 78)
( 8, 77)( 9, 91)( 10, 92)( 11, 90)( 12, 89)( 13, 95)( 14, 96)( 15, 94)
( 16, 93)( 17, 83)( 18, 84)( 19, 82)( 20, 81)( 21, 87)( 22, 88)( 23, 86)
( 24, 85)( 25, 99)( 26,100)( 27, 98)( 28, 97)( 29,103)( 30,104)( 31,102)
( 32,101)( 33,115)( 34,116)( 35,114)( 36,113)( 37,119)( 38,120)( 39,118)
( 40,117)( 41,107)( 42,108)( 43,106)( 44,105)( 45,111)( 46,112)( 47,110)
( 48,109)( 49,123)( 50,124)( 51,122)( 52,121)( 53,127)( 54,128)( 55,126)
( 56,125)( 57,139)( 58,140)( 59,138)( 60,137)( 61,143)( 62,144)( 63,142)
( 64,141)( 65,131)( 66,132)( 67,130)( 68,129)( 69,135)( 70,136)( 71,134)
( 72,133)(145,219)(146,220)(147,218)(148,217)(149,223)(150,224)(151,222)
(152,221)(153,235)(154,236)(155,234)(156,233)(157,239)(158,240)(159,238)
(160,237)(161,227)(162,228)(163,226)(164,225)(165,231)(166,232)(167,230)
(168,229)(169,243)(170,244)(171,242)(172,241)(173,247)(174,248)(175,246)
(176,245)(177,259)(178,260)(179,258)(180,257)(181,263)(182,264)(183,262)
(184,261)(185,251)(186,252)(187,250)(188,249)(189,255)(190,256)(191,254)
(192,253)(193,267)(194,268)(195,266)(196,265)(197,271)(198,272)(199,270)
(200,269)(201,283)(202,284)(203,282)(204,281)(205,287)(206,288)(207,286)
(208,285)(209,275)(210,276)(211,274)(212,273)(213,279)(214,280)(215,278)
(216,277);
s3 := Sym(288)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 25, 33)( 26, 34)( 27, 35)( 28, 36)( 29, 37)( 30, 38)( 31, 39)
( 32, 40)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 73, 81)( 74, 82)( 75, 83)( 76, 84)( 77, 85)( 78, 86)( 79, 87)
( 80, 88)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(121,129)(122,130)(123,131)(124,132)(125,133)(126,134)(127,135)
(128,136)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)
(152,160)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,183)
(176,184)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)
(200,208)(217,225)(218,226)(219,227)(220,228)(221,229)(222,230)(223,231)
(224,232)(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)
(248,256)(265,273)(266,274)(267,275)(268,276)(269,277)(270,278)(271,279)
(272,280);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope