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# Polytope of Type {6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*1152k
if this polytope has a name.
Group : SmallGroup(1152,156074)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 24, 288, 96
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,12}*576f
3-fold quotients : {6,8}*384g
4-fold quotients : {6,24}*288c, {6,12}*288b
6-fold quotients : {6,4}*192b
8-fold quotients : {6,12}*144c, {3,12}*144
12-fold quotients : {6,8}*96, {6,4}*96
16-fold quotients : {6,6}*72c
24-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
32-fold quotients : {3,6}*36
36-fold quotients : {2,8}*32
48-fold quotients : {3,4}*24, {6,2}*24
72-fold quotients : {2,4}*16
96-fold quotients : {3,2}*12
144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 25)( 14, 26)( 15, 28)
( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)( 23, 32)
( 24, 31)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)
( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)
( 59, 68)( 60, 67)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 97)
( 86, 98)( 87,100)( 88, 99)( 89,105)( 90,106)( 91,108)( 92,107)( 93,101)
( 94,102)( 95,104)( 96,103)(111,112)(113,117)(114,118)(115,120)(116,119)
(121,133)(122,134)(123,136)(124,135)(125,141)(126,142)(127,144)(128,143)
(129,137)(130,138)(131,140)(132,139)(147,148)(149,153)(150,154)(151,156)
(152,155)(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)(163,180)
(164,179)(165,173)(166,174)(167,176)(168,175)(183,184)(185,189)(186,190)
(187,192)(188,191)(193,205)(194,206)(195,208)(196,207)(197,213)(198,214)
(199,216)(200,215)(201,209)(202,210)(203,212)(204,211)(219,220)(221,225)
(222,226)(223,228)(224,227)(229,241)(230,242)(231,244)(232,243)(233,249)
(234,250)(235,252)(236,251)(237,245)(238,246)(239,248)(240,247)(255,256)
(257,261)(258,262)(259,264)(260,263)(265,277)(266,278)(267,280)(268,279)
(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)(276,283);;
s1 := (  1,161)(  2,164)(  3,163)(  4,162)(  5,157)(  6,160)(  7,159)(  8,158)
(  9,165)( 10,168)( 11,167)( 12,166)( 13,149)( 14,152)( 15,151)( 16,150)
( 17,145)( 18,148)( 19,147)( 20,146)( 21,153)( 22,156)( 23,155)( 24,154)
( 25,173)( 26,176)( 27,175)( 28,174)( 29,169)( 30,172)( 31,171)( 32,170)
( 33,177)( 34,180)( 35,179)( 36,178)( 37,197)( 38,200)( 39,199)( 40,198)
( 41,193)( 42,196)( 43,195)( 44,194)( 45,201)( 46,204)( 47,203)( 48,202)
( 49,185)( 50,188)( 51,187)( 52,186)( 53,181)( 54,184)( 55,183)( 56,182)
( 57,189)( 58,192)( 59,191)( 60,190)( 61,209)( 62,212)( 63,211)( 64,210)
( 65,205)( 66,208)( 67,207)( 68,206)( 69,213)( 70,216)( 71,215)( 72,214)
( 73,269)( 74,272)( 75,271)( 76,270)( 77,265)( 78,268)( 79,267)( 80,266)
( 81,273)( 82,276)( 83,275)( 84,274)( 85,257)( 86,260)( 87,259)( 88,258)
( 89,253)( 90,256)( 91,255)( 92,254)( 93,261)( 94,264)( 95,263)( 96,262)
( 97,281)( 98,284)( 99,283)(100,282)(101,277)(102,280)(103,279)(104,278)
(105,285)(106,288)(107,287)(108,286)(109,233)(110,236)(111,235)(112,234)
(113,229)(114,232)(115,231)(116,230)(117,237)(118,240)(119,239)(120,238)
(121,221)(122,224)(123,223)(124,222)(125,217)(126,220)(127,219)(128,218)
(129,225)(130,228)(131,227)(132,226)(133,245)(134,248)(135,247)(136,246)
(137,241)(138,244)(139,243)(140,242)(141,249)(142,252)(143,251)(144,250);;
s2 := (  1,  2)(  3,  4)(  5, 10)(  6,  9)(  7, 12)(  8, 11)( 13, 14)( 15, 16)
( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 26)( 27, 28)( 29, 34)( 30, 33)
( 31, 36)( 32, 35)( 37, 38)( 39, 40)( 41, 46)( 42, 45)( 43, 48)( 44, 47)
( 49, 50)( 51, 52)( 53, 58)( 54, 57)( 55, 60)( 56, 59)( 61, 62)( 63, 64)
( 65, 70)( 66, 69)( 67, 72)( 68, 71)( 73,110)( 74,109)( 75,112)( 76,111)
( 77,118)( 78,117)( 79,120)( 80,119)( 81,114)( 82,113)( 83,116)( 84,115)
( 85,122)( 86,121)( 87,124)( 88,123)( 89,130)( 90,129)( 91,132)( 92,131)
( 93,126)( 94,125)( 95,128)( 96,127)( 97,134)( 98,133)( 99,136)(100,135)
(101,142)(102,141)(103,144)(104,143)(105,138)(106,137)(107,140)(108,139)
(145,218)(146,217)(147,220)(148,219)(149,226)(150,225)(151,228)(152,227)
(153,222)(154,221)(155,224)(156,223)(157,230)(158,229)(159,232)(160,231)
(161,238)(162,237)(163,240)(164,239)(165,234)(166,233)(167,236)(168,235)
(169,242)(170,241)(171,244)(172,243)(173,250)(174,249)(175,252)(176,251)
(177,246)(178,245)(179,248)(180,247)(181,254)(182,253)(183,256)(184,255)
(185,262)(186,261)(187,264)(188,263)(189,258)(190,257)(191,260)(192,259)
(193,266)(194,265)(195,268)(196,267)(197,274)(198,273)(199,276)(200,275)
(201,270)(202,269)(203,272)(204,271)(205,278)(206,277)(207,280)(208,279)
(209,286)(210,285)(211,288)(212,287)(213,282)(214,281)(215,284)(216,283);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope