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# Polytope of Type {6,6,4}

Atlas Canonical Name : {6,6,4}*1152h
if this polytope has a name.
Group : SmallGroup(1152,157851)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 24, 72, 48, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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 : {6,3,4}*576
4-fold quotients : {6,6,4}*288f
8-fold quotients : {6,3,4}*144
12-fold quotients : {2,6,4}*96b
24-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 49, 97)( 50, 98)( 51,100)( 52, 99)
( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)
( 61,109)( 62,110)( 63,112)( 64,111)( 65,113)( 66,114)( 67,116)( 68,115)
( 69,117)( 70,118)( 71,120)( 72,119)( 73,121)( 74,122)( 75,124)( 76,123)
( 77,125)( 78,126)( 79,128)( 80,127)( 81,129)( 82,130)( 83,132)( 84,131)
( 85,133)( 86,134)( 87,136)( 88,135)( 89,137)( 90,138)( 91,140)( 92,139)
( 93,141)( 94,142)( 95,144)( 96,143)(147,148)(151,152)(155,156)(159,160)
(163,164)(167,168)(171,172)(175,176)(179,180)(183,184)(187,188)(191,192)
(193,241)(194,242)(195,244)(196,243)(197,245)(198,246)(199,248)(200,247)
(201,249)(202,250)(203,252)(204,251)(205,253)(206,254)(207,256)(208,255)
(209,257)(210,258)(211,260)(212,259)(213,261)(214,262)(215,264)(216,263)
(217,265)(218,266)(219,268)(220,267)(221,269)(222,270)(223,272)(224,271)
(225,273)(226,274)(227,276)(228,275)(229,277)(230,278)(231,280)(232,279)
(233,281)(234,282)(235,284)(236,283)(237,285)(238,286)(239,288)(240,287);;
s1 := (  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 53)(  6, 56)(  7, 55)(  8, 54)
(  9, 61)( 10, 64)( 11, 63)( 12, 62)( 13, 57)( 14, 60)( 15, 59)( 16, 58)
( 17, 81)( 18, 84)( 19, 83)( 20, 82)( 21, 85)( 22, 88)( 23, 87)( 24, 86)
( 25, 93)( 26, 96)( 27, 95)( 28, 94)( 29, 89)( 30, 92)( 31, 91)( 32, 90)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 37, 69)( 38, 72)( 39, 71)( 40, 70)
( 41, 77)( 42, 80)( 43, 79)( 44, 78)( 45, 73)( 46, 76)( 47, 75)( 48, 74)
( 98,100)(102,104)(105,109)(106,112)(107,111)(108,110)(113,129)(114,132)
(115,131)(116,130)(117,133)(118,136)(119,135)(120,134)(121,141)(122,144)
(123,143)(124,142)(125,137)(126,140)(127,139)(128,138)(145,193)(146,196)
(147,195)(148,194)(149,197)(150,200)(151,199)(152,198)(153,205)(154,208)
(155,207)(156,206)(157,201)(158,204)(159,203)(160,202)(161,225)(162,228)
(163,227)(164,226)(165,229)(166,232)(167,231)(168,230)(169,237)(170,240)
(171,239)(172,238)(173,233)(174,236)(175,235)(176,234)(177,209)(178,212)
(179,211)(180,210)(181,213)(182,216)(183,215)(184,214)(185,221)(186,224)
(187,223)(188,222)(189,217)(190,220)(191,219)(192,218)(242,244)(246,248)
(249,253)(250,256)(251,255)(252,254)(257,273)(258,276)(259,275)(260,274)
(261,277)(262,280)(263,279)(264,278)(265,285)(266,288)(267,287)(268,286)
(269,281)(270,284)(271,283)(272,282);;
s2 := (  1,162)(  2,161)(  3,163)(  4,164)(  5,174)(  6,173)(  7,175)(  8,176)
(  9,170)( 10,169)( 11,171)( 12,172)( 13,166)( 14,165)( 15,167)( 16,168)
( 17,146)( 18,145)( 19,147)( 20,148)( 21,158)( 22,157)( 23,159)( 24,160)
( 25,154)( 26,153)( 27,155)( 28,156)( 29,150)( 30,149)( 31,151)( 32,152)
( 33,178)( 34,177)( 35,179)( 36,180)( 37,190)( 38,189)( 39,191)( 40,192)
( 41,186)( 42,185)( 43,187)( 44,188)( 45,182)( 46,181)( 47,183)( 48,184)
( 49,258)( 50,257)( 51,259)( 52,260)( 53,270)( 54,269)( 55,271)( 56,272)
( 57,266)( 58,265)( 59,267)( 60,268)( 61,262)( 62,261)( 63,263)( 64,264)
( 65,242)( 66,241)( 67,243)( 68,244)( 69,254)( 70,253)( 71,255)( 72,256)
( 73,250)( 74,249)( 75,251)( 76,252)( 77,246)( 78,245)( 79,247)( 80,248)
( 81,274)( 82,273)( 83,275)( 84,276)( 85,286)( 86,285)( 87,287)( 88,288)
( 89,282)( 90,281)( 91,283)( 92,284)( 93,278)( 94,277)( 95,279)( 96,280)
( 97,210)( 98,209)( 99,211)(100,212)(101,222)(102,221)(103,223)(104,224)
(105,218)(106,217)(107,219)(108,220)(109,214)(110,213)(111,215)(112,216)
(113,194)(114,193)(115,195)(116,196)(117,206)(118,205)(119,207)(120,208)
(121,202)(122,201)(123,203)(124,204)(125,198)(126,197)(127,199)(128,200)
(129,226)(130,225)(131,227)(132,228)(133,238)(134,237)(135,239)(136,240)
(137,234)(138,233)(139,235)(140,236)(141,230)(142,229)(143,231)(144,232);;
s3 := (  1,149)(  2,150)(  3,151)(  4,152)(  5,145)(  6,146)(  7,147)(  8,148)
(  9,157)( 10,158)( 11,159)( 12,160)( 13,153)( 14,154)( 15,155)( 16,156)
( 17,165)( 18,166)( 19,167)( 20,168)( 21,161)( 22,162)( 23,163)( 24,164)
( 25,173)( 26,174)( 27,175)( 28,176)( 29,169)( 30,170)( 31,171)( 32,172)
( 33,181)( 34,182)( 35,183)( 36,184)( 37,177)( 38,178)( 39,179)( 40,180)
( 41,189)( 42,190)( 43,191)( 44,192)( 45,185)( 46,186)( 47,187)( 48,188)
( 49,197)( 50,198)( 51,199)( 52,200)( 53,193)( 54,194)( 55,195)( 56,196)
( 57,205)( 58,206)( 59,207)( 60,208)( 61,201)( 62,202)( 63,203)( 64,204)
( 65,213)( 66,214)( 67,215)( 68,216)( 69,209)( 70,210)( 71,211)( 72,212)
( 73,221)( 74,222)( 75,223)( 76,224)( 77,217)( 78,218)( 79,219)( 80,220)
( 81,229)( 82,230)( 83,231)( 84,232)( 85,225)( 86,226)( 87,227)( 88,228)
( 89,237)( 90,238)( 91,239)( 92,240)( 93,233)( 94,234)( 95,235)( 96,236)
( 97,245)( 98,246)( 99,247)(100,248)(101,241)(102,242)(103,243)(104,244)
(105,253)(106,254)(107,255)(108,256)(109,249)(110,250)(111,251)(112,252)
(113,261)(114,262)(115,263)(116,264)(117,257)(118,258)(119,259)(120,260)
(121,269)(122,270)(123,271)(124,272)(125,265)(126,266)(127,267)(128,268)
(129,277)(130,278)(131,279)(132,280)(133,273)(134,274)(135,275)(136,276)
(137,285)(138,286)(139,287)(140,288)(141,281)(142,282)(143,283)(144,284);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s1*s2*s3*s1*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope