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