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# Polytope of Type {8,12}

Atlas Canonical Name : {8,12}*576a
if this polytope has a name.
Group : SmallGroup(576,5307)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 24, 144, 36
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{8,12,2} of size 1152
Vertex Figure Of :
{2,8,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,6}*288, {4,12}*288
4-fold quotients : {4,6}*144
8-fold quotients : {4,6}*72
9-fold quotients : {8,4}*64a
18-fold quotients : {4,4}*32, {8,2}*32
36-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,12}*1152a, {8,24}*1152b, {8,24}*1152c, {16,12}*1152a, {16,12}*1152b
3-fold covers : {8,12}*1728a, {24,12}*1728g, {24,12}*1728h, {8,12}*1728g, {24,12}*1728s, {24,12}*1728u
Permutation Representation (GAP) :
```s0 := (  1,145)(  2,148)(  3,151)(  4,146)(  5,149)(  6,152)(  7,147)(  8,150)
(  9,153)( 10,154)( 11,157)( 12,160)( 13,155)( 14,158)( 15,161)( 16,156)
( 17,159)( 18,162)( 19,163)( 20,166)( 21,169)( 22,164)( 23,167)( 24,170)
( 25,165)( 26,168)( 27,171)( 28,172)( 29,175)( 30,178)( 31,173)( 32,176)
( 33,179)( 34,174)( 35,177)( 36,180)( 37,190)( 38,193)( 39,196)( 40,191)
( 41,194)( 42,197)( 43,192)( 44,195)( 45,198)( 46,181)( 47,184)( 48,187)
( 49,182)( 50,185)( 51,188)( 52,183)( 53,186)( 54,189)( 55,208)( 56,211)
( 57,214)( 58,209)( 59,212)( 60,215)( 61,210)( 62,213)( 63,216)( 64,199)
( 65,202)( 66,205)( 67,200)( 68,203)( 69,206)( 70,201)( 71,204)( 72,207)
( 73,226)( 74,229)( 75,232)( 76,227)( 77,230)( 78,233)( 79,228)( 80,231)
( 81,234)( 82,217)( 83,220)( 84,223)( 85,218)( 86,221)( 87,224)( 88,219)
( 89,222)( 90,225)( 91,244)( 92,247)( 93,250)( 94,245)( 95,248)( 96,251)
( 97,246)( 98,249)( 99,252)(100,235)(101,238)(102,241)(103,236)(104,239)
(105,242)(106,237)(107,240)(108,243)(109,253)(110,256)(111,259)(112,254)
(113,257)(114,260)(115,255)(116,258)(117,261)(118,262)(119,265)(120,268)
(121,263)(122,266)(123,269)(124,264)(125,267)(126,270)(127,271)(128,274)
(129,277)(130,272)(131,275)(132,278)(133,273)(134,276)(135,279)(136,280)
(137,283)(138,286)(139,281)(140,284)(141,287)(142,282)(143,285)(144,288);;
s1 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)
( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 55, 64)( 56, 65)( 57, 66)
( 58, 70)( 59, 71)( 60, 72)( 61, 67)( 62, 68)( 63, 69)( 73, 91)( 74, 92)
( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)( 82,100)
( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)( 88,103)( 89,104)( 90,105)
(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)
(117,141)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)
(125,131)(126,132)(145,181)(146,182)(147,183)(148,187)(149,188)(150,189)
(151,184)(152,185)(153,186)(154,190)(155,191)(156,192)(157,196)(158,197)
(159,198)(160,193)(161,194)(162,195)(163,199)(164,200)(165,201)(166,205)
(167,206)(168,207)(169,202)(170,203)(171,204)(172,208)(173,209)(174,210)
(175,214)(176,215)(177,216)(178,211)(179,212)(180,213)(217,271)(218,272)
(219,273)(220,277)(221,278)(222,279)(223,274)(224,275)(225,276)(226,280)
(227,281)(228,282)(229,286)(230,287)(231,288)(232,283)(233,284)(234,285)
(235,253)(236,254)(237,255)(238,259)(239,260)(240,261)(241,256)(242,257)
(243,258)(244,262)(245,263)(246,264)(247,268)(248,269)(249,270)(250,265)
(251,266)(252,267);;
s2 := (  1,113)(  2,112)(  3,114)(  4,110)(  5,109)(  6,111)(  7,116)(  8,115)
(  9,117)( 10,122)( 11,121)( 12,123)( 13,119)( 14,118)( 15,120)( 16,125)
( 17,124)( 18,126)( 19,131)( 20,130)( 21,132)( 22,128)( 23,127)( 24,129)
( 25,134)( 26,133)( 27,135)( 28,140)( 29,139)( 30,141)( 31,137)( 32,136)
( 33,138)( 34,143)( 35,142)( 36,144)( 37, 86)( 38, 85)( 39, 87)( 40, 83)
( 41, 82)( 42, 84)( 43, 89)( 44, 88)( 45, 90)( 46, 77)( 47, 76)( 48, 78)
( 49, 74)( 50, 73)( 51, 75)( 52, 80)( 53, 79)( 54, 81)( 55,104)( 56,103)
( 57,105)( 58,101)( 59,100)( 60,102)( 61,107)( 62,106)( 63,108)( 64, 95)
( 65, 94)( 66, 96)( 67, 92)( 68, 91)( 69, 93)( 70, 98)( 71, 97)( 72, 99)
(145,257)(146,256)(147,258)(148,254)(149,253)(150,255)(151,260)(152,259)
(153,261)(154,266)(155,265)(156,267)(157,263)(158,262)(159,264)(160,269)
(161,268)(162,270)(163,275)(164,274)(165,276)(166,272)(167,271)(168,273)
(169,278)(170,277)(171,279)(172,284)(173,283)(174,285)(175,281)(176,280)
(177,282)(178,287)(179,286)(180,288)(181,230)(182,229)(183,231)(184,227)
(185,226)(186,228)(187,233)(188,232)(189,234)(190,221)(191,220)(192,222)
(193,218)(194,217)(195,219)(196,224)(197,223)(198,225)(199,248)(200,247)
(201,249)(202,245)(203,244)(204,246)(205,251)(206,250)(207,252)(208,239)
(209,238)(210,240)(211,236)(212,235)(213,237)(214,242)(215,241)(216,243);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope