Polytope of Type {4,12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,6}*1728s
if this polytope has a name.
Group : SmallGroup(1728,47870)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 8, 72, 108, 9
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {4,12,6}*864d
4-fold quotients : {2,12,6}*432c
12-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)( 8,114)
( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)
( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)
( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)
( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)
( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214);;
s1 := ( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)
( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)
( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 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, 85)
( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)
( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)(122,134)
(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)
(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)
(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)(158,206)
(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)(166,214)
(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)(174,198)
(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);;
s2 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 61)( 6, 64)( 7, 63)( 8, 62)
( 9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 45)( 14, 48)( 15, 47)( 16, 46)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 41)( 26, 44)( 27, 43)( 28, 42)( 29, 65)( 30, 68)( 31, 67)( 32, 66)
( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 74, 76)( 77, 97)( 78,100)( 79, 99)
( 80, 98)( 81, 85)( 82, 88)( 83, 87)( 84, 86)( 89,105)( 90,108)( 91,107)
( 92,106)( 94, 96)(102,104)(109,145)(110,148)(111,147)(112,146)(113,169)
(114,172)(115,171)(116,170)(117,157)(118,160)(119,159)(120,158)(121,153)
(122,156)(123,155)(124,154)(125,177)(126,180)(127,179)(128,178)(129,165)
(130,168)(131,167)(132,166)(133,149)(134,152)(135,151)(136,150)(137,173)
(138,176)(139,175)(140,174)(141,161)(142,164)(143,163)(144,162)(182,184)
(185,205)(186,208)(187,207)(188,206)(189,193)(190,196)(191,195)(192,194)
(197,213)(198,216)(199,215)(200,214)(202,204)(210,212);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 29)( 14, 30)( 15, 31)( 16, 32)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)( 24, 36)
( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 65)( 50, 66)( 51, 67)( 52, 68)
( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85,101)( 86,102)( 87,103)( 88,104)
( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)( 96,108)
(109,113)(110,114)(111,115)(112,116)(121,137)(122,138)(123,139)(124,140)
(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)(132,144)
(145,149)(146,150)(147,151)(148,152)(157,173)(158,174)(159,175)(160,176)
(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)(168,180)
(181,185)(182,186)(183,187)(184,188)(193,209)(194,210)(195,211)(196,212)
(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)(204,216);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)
( 8,114)( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)
( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)
( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)
( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)
( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)
( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)
( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)
( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)
( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)
( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)
(104,210)(105,215)(106,216)(107,213)(108,214);
s1 := Sym(216)!( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)
( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)
( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)
( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 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, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)
( 69, 93)( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)
(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)
(130,142)(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)
(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)
(158,206)(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)
(166,214)(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)
(174,198)(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);
s2 := Sym(216)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 61)( 6, 64)( 7, 63)
( 8, 62)( 9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 45)( 14, 48)( 15, 47)
( 16, 46)( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 57)( 22, 60)( 23, 59)
( 24, 58)( 25, 41)( 26, 44)( 27, 43)( 28, 42)( 29, 65)( 30, 68)( 31, 67)
( 32, 66)( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 74, 76)( 77, 97)( 78,100)
( 79, 99)( 80, 98)( 81, 85)( 82, 88)( 83, 87)( 84, 86)( 89,105)( 90,108)
( 91,107)( 92,106)( 94, 96)(102,104)(109,145)(110,148)(111,147)(112,146)
(113,169)(114,172)(115,171)(116,170)(117,157)(118,160)(119,159)(120,158)
(121,153)(122,156)(123,155)(124,154)(125,177)(126,180)(127,179)(128,178)
(129,165)(130,168)(131,167)(132,166)(133,149)(134,152)(135,151)(136,150)
(137,173)(138,176)(139,175)(140,174)(141,161)(142,164)(143,163)(144,162)
(182,184)(185,205)(186,208)(187,207)(188,206)(189,193)(190,196)(191,195)
(192,194)(197,213)(198,216)(199,215)(200,214)(202,204)(210,212);
s3 := Sym(216)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 29)( 14, 30)( 15, 31)
( 16, 32)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)
( 24, 36)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 65)( 50, 66)( 51, 67)
( 52, 68)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)
( 60, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85,101)( 86,102)( 87,103)
( 88,104)( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)
( 96,108)(109,113)(110,114)(111,115)(112,116)(121,137)(122,138)(123,139)
(124,140)(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)
(132,144)(145,149)(146,150)(147,151)(148,152)(157,173)(158,174)(159,175)
(160,176)(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)
(168,180)(181,185)(182,186)(183,187)(184,188)(193,209)(194,210)(195,211)
(196,212)(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)
(204,216);
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,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3 >;
References : None.
to this polytope