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# Polytope of Type {12,6,4}

Atlas Canonical Name : {12,6,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,157550)
Rank : 4
Schlafli Type : {12,6,4}
Number of vertices, edges, etc : 12, 72, 24, 8
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 : {12,6,4}*576d, {6,6,4}*576a
3-fold quotients : {4,6,4}*384a
4-fold quotients : {12,6,2}*288a, {6,6,4}*288d
6-fold quotients : {4,6,4}*192b, {2,6,4}*192
8-fold quotients : {6,6,2}*144a
12-fold quotients : {12,2,2}*96, {4,6,2}*96a, {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
24-fold quotients : {2,3,4}*48, {2,6,2}*48, {6,2,2}*48
36-fold quotients : {4,2,2}*32
48-fold quotients : {2,3,2}*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 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 73,109)( 74,110)( 75,111)( 76,112)( 77,117)( 78,118)( 79,119)( 80,120)
( 81,113)( 82,114)( 83,115)( 84,116)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,125)( 94,126)( 95,127)( 96,128)
( 97,133)( 98,134)( 99,135)(100,136)(101,141)(102,142)(103,143)(104,144)
(105,137)(106,138)(107,139)(108,140)(149,153)(150,154)(151,155)(152,156)
(161,165)(162,166)(163,167)(164,168)(173,177)(174,178)(175,179)(176,180)
(185,189)(186,190)(187,191)(188,192)(197,201)(198,202)(199,203)(200,204)
(209,213)(210,214)(211,215)(212,216)(217,253)(218,254)(219,255)(220,256)
(221,261)(222,262)(223,263)(224,264)(225,257)(226,258)(227,259)(228,260)
(229,265)(230,266)(231,267)(232,268)(233,273)(234,274)(235,275)(236,276)
(237,269)(238,270)(239,271)(240,272)(241,277)(242,278)(243,279)(244,280)
(245,285)(246,286)(247,287)(248,288)(249,281)(250,282)(251,283)(252,284);;
s1 := (  1, 77)(  2, 78)(  3, 80)(  4, 79)(  5, 73)(  6, 74)(  7, 76)(  8, 75)
(  9, 81)( 10, 82)( 11, 84)( 12, 83)( 13,101)( 14,102)( 15,104)( 16,103)
( 17, 97)( 18, 98)( 19,100)( 20, 99)( 21,105)( 22,106)( 23,108)( 24,107)
( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 85)( 30, 86)( 31, 88)( 32, 87)
( 33, 93)( 34, 94)( 35, 96)( 36, 95)( 37,113)( 38,114)( 39,116)( 40,115)
( 41,109)( 42,110)( 43,112)( 44,111)( 45,117)( 46,118)( 47,120)( 48,119)
( 49,137)( 50,138)( 51,140)( 52,139)( 53,133)( 54,134)( 55,136)( 56,135)
( 57,141)( 58,142)( 59,144)( 60,143)( 61,125)( 62,126)( 63,128)( 64,127)
( 65,121)( 66,122)( 67,124)( 68,123)( 69,129)( 70,130)( 71,132)( 72,131)
(145,221)(146,222)(147,224)(148,223)(149,217)(150,218)(151,220)(152,219)
(153,225)(154,226)(155,228)(156,227)(157,245)(158,246)(159,248)(160,247)
(161,241)(162,242)(163,244)(164,243)(165,249)(166,250)(167,252)(168,251)
(169,233)(170,234)(171,236)(172,235)(173,229)(174,230)(175,232)(176,231)
(177,237)(178,238)(179,240)(180,239)(181,257)(182,258)(183,260)(184,259)
(185,253)(186,254)(187,256)(188,255)(189,261)(190,262)(191,264)(192,263)
(193,281)(194,282)(195,284)(196,283)(197,277)(198,278)(199,280)(200,279)
(201,285)(202,286)(203,288)(204,287)(205,269)(206,270)(207,272)(208,271)
(209,265)(210,266)(211,268)(212,267)(213,273)(214,274)(215,276)(216,275);;
s2 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 17)(  6, 20)(  7, 19)(  8, 18)
(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)
( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)
( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)( 74, 88)
( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)( 82, 96)
( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)(111,123)
(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)(119,131)
(120,130)(134,136)(138,140)(142,144)(145,157)(146,160)(147,159)(148,158)
(149,161)(150,164)(151,163)(152,162)(153,165)(154,168)(155,167)(156,166)
(170,172)(174,176)(178,180)(181,193)(182,196)(183,195)(184,194)(185,197)
(186,200)(187,199)(188,198)(189,201)(190,204)(191,203)(192,202)(206,208)
(210,212)(214,216)(217,229)(218,232)(219,231)(220,230)(221,233)(222,236)
(223,235)(224,234)(225,237)(226,240)(227,239)(228,238)(242,244)(246,248)
(250,252)(253,265)(254,268)(255,267)(256,266)(257,269)(258,272)(259,271)
(260,270)(261,273)(262,276)(263,275)(264,274)(278,280)(282,284)(286,288);;
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*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope