Polytope of Type {6,4,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,12703)
Rank : 4
Schlafli Type : {6,4,8}
Number of vertices, edges, etc : 27, 54, 72, 8
Order of s0s1s2s3 : 24
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 : {6,4,4}*864a
3-fold quotients : {6,4,8}*576
4-fold quotients : {6,4,2}*432
6-fold quotients : {6,4,4}*288
12-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 9)( 5, 7)( 6, 8)( 10, 21)( 11, 19)( 12, 20)( 13, 26)( 14, 27)
( 15, 25)( 16, 22)( 17, 23)( 18, 24)( 31, 36)( 32, 34)( 33, 35)( 37, 48)
( 38, 46)( 39, 47)( 40, 53)( 41, 54)( 42, 52)( 43, 49)( 44, 50)( 45, 51)
( 58, 63)( 59, 61)( 60, 62)( 64, 75)( 65, 73)( 66, 74)( 67, 80)( 68, 81)
( 69, 79)( 70, 76)( 71, 77)( 72, 78)( 85, 90)( 86, 88)( 87, 89)( 91,102)
( 92,100)( 93,101)( 94,107)( 95,108)( 96,106)( 97,103)( 98,104)( 99,105)
(112,117)(113,115)(114,116)(118,129)(119,127)(120,128)(121,134)(122,135)
(123,133)(124,130)(125,131)(126,132)(139,144)(140,142)(141,143)(145,156)
(146,154)(147,155)(148,161)(149,162)(150,160)(151,157)(152,158)(153,159)
(166,171)(167,169)(168,170)(172,183)(173,181)(174,182)(175,188)(176,189)
(177,187)(178,184)(179,185)(180,186)(193,198)(194,196)(195,197)(199,210)
(200,208)(201,209)(202,215)(203,216)(204,214)(205,211)(206,212)(207,213);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 5)( 7, 26)( 8, 25)( 9, 27)( 13, 23)
( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 31, 32)
( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)( 47, 48)
( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)( 67, 77)
( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)( 85, 86)
( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)(101,102)
(109,118)(110,120)(111,119)(112,113)(115,134)(116,133)(117,135)(121,131)
(122,130)(123,132)(124,125)(128,129)(136,145)(137,147)(138,146)(139,140)
(142,161)(143,160)(144,162)(148,158)(149,157)(150,159)(151,152)(155,156)
(163,172)(164,174)(165,173)(166,167)(169,188)(170,187)(171,189)(175,185)
(176,184)(177,186)(178,179)(182,183)(190,199)(191,201)(192,200)(193,194)
(196,215)(197,214)(198,216)(202,212)(203,211)(204,213)(205,206)(209,210);;
s2 := ( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)
( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)
( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)
( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)
( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)
( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)
( 79, 91)( 80, 93)( 81, 92)(109,163)(110,165)(111,164)(112,167)(113,166)
(114,168)(115,171)(116,170)(117,169)(118,187)(119,189)(120,188)(121,182)
(122,181)(123,183)(124,186)(125,185)(126,184)(127,176)(128,175)(129,177)
(130,180)(131,179)(132,178)(133,172)(134,174)(135,173)(136,190)(137,192)
(138,191)(139,194)(140,193)(141,195)(142,198)(143,197)(144,196)(145,214)
(146,216)(147,215)(148,209)(149,208)(150,210)(151,213)(152,212)(153,211)
(154,203)(155,202)(156,204)(157,207)(158,206)(159,205)(160,199)(161,201)
(162,200);;
s3 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 4, 9)( 5, 7)( 6, 8)( 10, 21)( 11, 19)( 12, 20)( 13, 26)
( 14, 27)( 15, 25)( 16, 22)( 17, 23)( 18, 24)( 31, 36)( 32, 34)( 33, 35)
( 37, 48)( 38, 46)( 39, 47)( 40, 53)( 41, 54)( 42, 52)( 43, 49)( 44, 50)
( 45, 51)( 58, 63)( 59, 61)( 60, 62)( 64, 75)( 65, 73)( 66, 74)( 67, 80)
( 68, 81)( 69, 79)( 70, 76)( 71, 77)( 72, 78)( 85, 90)( 86, 88)( 87, 89)
( 91,102)( 92,100)( 93,101)( 94,107)( 95,108)( 96,106)( 97,103)( 98,104)
( 99,105)(112,117)(113,115)(114,116)(118,129)(119,127)(120,128)(121,134)
(122,135)(123,133)(124,130)(125,131)(126,132)(139,144)(140,142)(141,143)
(145,156)(146,154)(147,155)(148,161)(149,162)(150,160)(151,157)(152,158)
(153,159)(166,171)(167,169)(168,170)(172,183)(173,181)(174,182)(175,188)
(176,189)(177,187)(178,184)(179,185)(180,186)(193,198)(194,196)(195,197)
(199,210)(200,208)(201,209)(202,215)(203,216)(204,214)(205,211)(206,212)
(207,213);
s1 := Sym(216)!( 1, 10)( 2, 12)( 3, 11)( 4, 5)( 7, 26)( 8, 25)( 9, 27)
( 13, 23)( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)
( 31, 32)( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)
( 47, 48)( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)
( 67, 77)( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)
( 85, 86)( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)
(101,102)(109,118)(110,120)(111,119)(112,113)(115,134)(116,133)(117,135)
(121,131)(122,130)(123,132)(124,125)(128,129)(136,145)(137,147)(138,146)
(139,140)(142,161)(143,160)(144,162)(148,158)(149,157)(150,159)(151,152)
(155,156)(163,172)(164,174)(165,173)(166,167)(169,188)(170,187)(171,189)
(175,185)(176,184)(177,186)(178,179)(182,183)(190,199)(191,201)(192,200)
(193,194)(196,215)(197,214)(198,216)(202,212)(203,211)(204,213)(205,206)
(209,210);
s2 := Sym(216)!( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)
( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)
( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)
( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)
( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)
( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)
( 78, 97)( 79, 91)( 80, 93)( 81, 92)(109,163)(110,165)(111,164)(112,167)
(113,166)(114,168)(115,171)(116,170)(117,169)(118,187)(119,189)(120,188)
(121,182)(122,181)(123,183)(124,186)(125,185)(126,184)(127,176)(128,175)
(129,177)(130,180)(131,179)(132,178)(133,172)(134,174)(135,173)(136,190)
(137,192)(138,191)(139,194)(140,193)(141,195)(142,198)(143,197)(144,196)
(145,214)(146,216)(147,215)(148,209)(149,208)(150,210)(151,213)(152,212)
(153,211)(154,203)(155,202)(156,204)(157,207)(158,206)(159,205)(160,199)
(161,201)(162,200);
s3 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
poly := sub<Sym(216)|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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope