Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,12}

Atlas Canonical Name {4,6,12}*1152f

Overview

Group
SmallGroup(1152,157864)
Rank
4
Schläfli Type
{4,6,12}
Vertices, edges, …
4, 24, 72, 24
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

12-fold

24-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s2*s3*s2*s1*(s3*s2)^2*s3> of order 2

12 facets

4 vertex figures

P/N, where N=<s1*s2*s1*s3*s2*s1*(s2*s3)^2*s2> of order 2

12 facets

4 vertex figures

P/N, where N=<s1*s2*s3*s2*s1*(s3*s2)^2> of order 2

12 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)(248,256)(257,265)(258,266)(259,267)(260,268)(261,269)(262,270)(263,271)(264,272)(273,281)(274,282)(275,283)(276,284)(277,285)(278,286)(279,287)(280,288);;
s1 := (  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);;
s2 := (  1,209)(  2,212)(  3,211)(  4,210)(  5,221)(  6,224)(  7,223)(  8,222)(  9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)( 16,214)( 17,193)( 18,196)( 19,195)( 20,194)( 21,205)( 22,208)( 23,207)( 24,206)( 25,201)( 26,204)( 27,203)( 28,202)( 29,197)( 30,200)( 31,199)( 32,198)( 33,225)( 34,228)( 35,227)( 36,226)( 37,237)( 38,240)( 39,239)( 40,238)( 41,233)( 42,236)( 43,235)( 44,234)( 45,229)( 46,232)( 47,231)( 48,230)( 49,161)( 50,164)( 51,163)( 52,162)( 53,173)( 54,176)( 55,175)( 56,174)( 57,169)( 58,172)( 59,171)( 60,170)( 61,165)( 62,168)( 63,167)( 64,166)( 65,145)( 66,148)( 67,147)( 68,146)( 69,157)( 70,160)( 71,159)( 72,158)( 73,153)( 74,156)( 75,155)( 76,154)( 77,149)( 78,152)( 79,151)( 80,150)( 81,177)( 82,180)( 83,179)( 84,178)( 85,189)( 86,192)( 87,191)( 88,190)( 89,185)( 90,188)( 91,187)( 92,186)( 93,181)( 94,184)( 95,183)( 96,182)( 97,257)( 98,260)( 99,259)(100,258)(101,269)(102,272)(103,271)(104,270)(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)(112,262)(113,241)(114,244)(115,243)(116,242)(117,253)(118,256)(119,255)(120,254)(121,249)(122,252)(123,251)(124,250)(125,245)(126,248)(127,247)(128,246)(129,273)(130,276)(131,275)(132,274)(133,285)(134,288)(135,287)(136,286)(137,281)(138,284)(139,283)(140,282)(141,277)(142,280)(143,279)(144,278);;
s3 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 40)( 24, 39)( 25, 42)( 26, 41)( 27, 44)( 28, 43)( 29, 46)( 30, 45)( 31, 48)( 32, 47)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 82)( 66, 81)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 88)( 72, 87)( 73, 90)( 74, 89)( 75, 92)( 76, 91)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,130)(114,129)(115,132)(116,131)(117,134)(118,133)(119,136)(120,135)(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)(128,143)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,178)(162,177)(163,180)(164,179)(165,182)(166,181)(167,184)(168,183)(169,186)(170,185)(171,188)(172,187)(173,190)(174,189)(175,192)(176,191)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,226)(210,225)(211,228)(212,227)(213,230)(214,229)(215,232)(216,231)(217,234)(218,233)(219,236)(220,235)(221,238)(222,237)(223,240)(224,239)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,274)(258,273)(259,276)(260,275)(261,278)(262,277)(263,280)(264,279)(265,282)(266,281)(267,284)(268,283)(269,286)(270,285)(271,288)(272,287);;
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*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(288)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)(248,256)(257,265)(258,266)(259,267)(260,268)(261,269)(262,270)(263,271)(264,272)(273,281)(274,282)(275,283)(276,284)(277,285)(278,286)(279,287)(280,288);
s1 := 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);
s2 := Sym(288)!(  1,209)(  2,212)(  3,211)(  4,210)(  5,221)(  6,224)(  7,223)(  8,222)(  9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)( 16,214)( 17,193)( 18,196)( 19,195)( 20,194)( 21,205)( 22,208)( 23,207)( 24,206)( 25,201)( 26,204)( 27,203)( 28,202)( 29,197)( 30,200)( 31,199)( 32,198)( 33,225)( 34,228)( 35,227)( 36,226)( 37,237)( 38,240)( 39,239)( 40,238)( 41,233)( 42,236)( 43,235)( 44,234)( 45,229)( 46,232)( 47,231)( 48,230)( 49,161)( 50,164)( 51,163)( 52,162)( 53,173)( 54,176)( 55,175)( 56,174)( 57,169)( 58,172)( 59,171)( 60,170)( 61,165)( 62,168)( 63,167)( 64,166)( 65,145)( 66,148)( 67,147)( 68,146)( 69,157)( 70,160)( 71,159)( 72,158)( 73,153)( 74,156)( 75,155)( 76,154)( 77,149)( 78,152)( 79,151)( 80,150)( 81,177)( 82,180)( 83,179)( 84,178)( 85,189)( 86,192)( 87,191)( 88,190)( 89,185)( 90,188)( 91,187)( 92,186)( 93,181)( 94,184)( 95,183)( 96,182)( 97,257)( 98,260)( 99,259)(100,258)(101,269)(102,272)(103,271)(104,270)(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)(112,262)(113,241)(114,244)(115,243)(116,242)(117,253)(118,256)(119,255)(120,254)(121,249)(122,252)(123,251)(124,250)(125,245)(126,248)(127,247)(128,246)(129,273)(130,276)(131,275)(132,274)(133,285)(134,288)(135,287)(136,286)(137,281)(138,284)(139,283)(140,282)(141,277)(142,280)(143,279)(144,278);
s3 := Sym(288)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 40)( 24, 39)( 25, 42)( 26, 41)( 27, 44)( 28, 43)( 29, 46)( 30, 45)( 31, 48)( 32, 47)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 82)( 66, 81)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 88)( 72, 87)( 73, 90)( 74, 89)( 75, 92)( 76, 91)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,130)(114,129)(115,132)(116,131)(117,134)(118,133)(119,136)(120,135)(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)(128,143)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,178)(162,177)(163,180)(164,179)(165,182)(166,181)(167,184)(168,183)(169,186)(170,185)(171,188)(172,187)(173,190)(174,189)(175,192)(176,191)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,226)(210,225)(211,228)(212,227)(213,230)(214,229)(215,232)(216,231)(217,234)(218,233)(219,236)(220,235)(221,238)(222,237)(223,240)(224,239)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,274)(258,273)(259,276)(260,275)(261,278)(262,277)(263,280)(264,279)(265,282)(266,281)(267,284)(268,283)(269,286)(270,285)(271,288)(272,287);
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*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2 >;

References

None.

to this polytope.