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# Polytope of Type {8,6,6}

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

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

```
References : None.
to this polytope