Polytope of Type {12,6,6,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,6,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {12,6,6,2}
Number of vertices, edges, etc : 12, 36, 18, 6, 2
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {12,6,3,2}*864a, {6,6,6,2}*864a
3-fold quotients : {12,2,6,2}*576
4-fold quotients : {3,6,6,2}*432a, {6,6,3,2}*432a
6-fold quotients : {12,2,3,2}*288, {6,2,6,2}*288
8-fold quotients : {3,6,3,2}*216
9-fold quotients : {12,2,2,2}*192, {4,2,6,2}*192
12-fold quotients : {3,2,6,2}*144, {6,2,3,2}*144
18-fold quotients : {4,2,3,2}*96, {2,2,6,2}*96, {6,2,2,2}*96
24-fold quotients : {3,2,3,2}*72
27-fold quotients : {4,2,2,2}*64
36-fold quotients : {2,2,3,2}*48, {3,2,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)
( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)
( 79, 97)( 80, 99)( 81, 98)(110,111)(113,114)(116,117)(118,127)(119,129)
(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)
(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)
(151,160)(152,162)(153,161)(163,190)(164,192)(165,191)(166,193)(167,195)
(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)(175,211)
(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)(183,200)
(184,202)(185,204)(186,203)(187,205)(188,207)(189,206);;
s1 := ( 1, 64)( 2, 66)( 3, 65)( 4, 68)( 5, 67)( 6, 69)( 7, 72)( 8, 71)
( 9, 70)( 10, 55)( 11, 57)( 12, 56)( 13, 59)( 14, 58)( 15, 60)( 16, 63)
( 17, 62)( 18, 61)( 19, 73)( 20, 75)( 21, 74)( 22, 77)( 23, 76)( 24, 78)
( 25, 81)( 26, 80)( 27, 79)( 28, 91)( 29, 93)( 30, 92)( 31, 95)( 32, 94)
( 33, 96)( 34, 99)( 35, 98)( 36, 97)( 37, 82)( 38, 84)( 39, 83)( 40, 86)
( 41, 85)( 42, 87)( 43, 90)( 44, 89)( 45, 88)( 46,100)( 47,102)( 48,101)
( 49,104)( 50,103)( 51,105)( 52,108)( 53,107)( 54,106)(109,172)(110,174)
(111,173)(112,176)(113,175)(114,177)(115,180)(116,179)(117,178)(118,163)
(119,165)(120,164)(121,167)(122,166)(123,168)(124,171)(125,170)(126,169)
(127,181)(128,183)(129,182)(130,185)(131,184)(132,186)(133,189)(134,188)
(135,187)(136,199)(137,201)(138,200)(139,203)(140,202)(141,204)(142,207)
(143,206)(144,205)(145,190)(146,192)(147,191)(148,194)(149,193)(150,195)
(151,198)(152,197)(153,196)(154,208)(155,210)(156,209)(157,212)(158,211)
(159,213)(160,216)(161,215)(162,214);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)
( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108)
(109,112)(110,114)(111,113)(116,117)(118,121)(119,123)(120,122)(125,126)
(127,130)(128,132)(129,131)(134,135)(136,139)(137,141)(138,140)(143,144)
(145,148)(146,150)(147,149)(152,153)(154,157)(155,159)(156,158)(161,162)
(163,166)(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)(179,180)
(181,184)(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)(197,198)
(199,202)(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)(215,216);;
s3 := ( 1,109)( 2,111)( 3,110)( 4,115)( 5,117)( 6,116)( 7,112)( 8,114)
( 9,113)( 10,118)( 11,120)( 12,119)( 13,124)( 14,126)( 15,125)( 16,121)
( 17,123)( 18,122)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)( 24,134)
( 25,130)( 26,132)( 27,131)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)
( 33,143)( 34,139)( 35,141)( 36,140)( 37,145)( 38,147)( 39,146)( 40,151)
( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,154)( 47,156)( 48,155)
( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,163)( 56,165)
( 57,164)( 58,169)( 59,171)( 60,170)( 61,166)( 62,168)( 63,167)( 64,172)
( 65,174)( 66,173)( 67,178)( 68,180)( 69,179)( 70,175)( 71,177)( 72,176)
( 73,181)( 74,183)( 75,182)( 76,187)( 77,189)( 78,188)( 79,184)( 80,186)
( 81,185)( 82,190)( 83,192)( 84,191)( 85,196)( 86,198)( 87,197)( 88,193)
( 89,195)( 90,194)( 91,199)( 92,201)( 93,200)( 94,205)( 95,207)( 96,206)
( 97,202)( 98,204)( 99,203)(100,208)(101,210)(102,209)(103,214)(104,216)
(105,215)(106,211)(107,213)(108,212);;
s4 := (217,218);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)
( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)
( 78, 95)( 79, 97)( 80, 99)( 81, 98)(110,111)(113,114)(116,117)(118,127)
(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)
(137,138)(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)
(150,158)(151,160)(152,162)(153,161)(163,190)(164,192)(165,191)(166,193)
(167,195)(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)
(175,211)(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)
(183,200)(184,202)(185,204)(186,203)(187,205)(188,207)(189,206);
s1 := Sym(218)!( 1, 64)( 2, 66)( 3, 65)( 4, 68)( 5, 67)( 6, 69)( 7, 72)
( 8, 71)( 9, 70)( 10, 55)( 11, 57)( 12, 56)( 13, 59)( 14, 58)( 15, 60)
( 16, 63)( 17, 62)( 18, 61)( 19, 73)( 20, 75)( 21, 74)( 22, 77)( 23, 76)
( 24, 78)( 25, 81)( 26, 80)( 27, 79)( 28, 91)( 29, 93)( 30, 92)( 31, 95)
( 32, 94)( 33, 96)( 34, 99)( 35, 98)( 36, 97)( 37, 82)( 38, 84)( 39, 83)
( 40, 86)( 41, 85)( 42, 87)( 43, 90)( 44, 89)( 45, 88)( 46,100)( 47,102)
( 48,101)( 49,104)( 50,103)( 51,105)( 52,108)( 53,107)( 54,106)(109,172)
(110,174)(111,173)(112,176)(113,175)(114,177)(115,180)(116,179)(117,178)
(118,163)(119,165)(120,164)(121,167)(122,166)(123,168)(124,171)(125,170)
(126,169)(127,181)(128,183)(129,182)(130,185)(131,184)(132,186)(133,189)
(134,188)(135,187)(136,199)(137,201)(138,200)(139,203)(140,202)(141,204)
(142,207)(143,206)(144,205)(145,190)(146,192)(147,191)(148,194)(149,193)
(150,195)(151,198)(152,197)(153,196)(154,208)(155,210)(156,209)(157,212)
(158,211)(159,213)(160,216)(161,215)(162,214);
s2 := Sym(218)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)
( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)
( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)
( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)
(107,108)(109,112)(110,114)(111,113)(116,117)(118,121)(119,123)(120,122)
(125,126)(127,130)(128,132)(129,131)(134,135)(136,139)(137,141)(138,140)
(143,144)(145,148)(146,150)(147,149)(152,153)(154,157)(155,159)(156,158)
(161,162)(163,166)(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)
(179,180)(181,184)(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)
(197,198)(199,202)(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)
(215,216);
s3 := Sym(218)!( 1,109)( 2,111)( 3,110)( 4,115)( 5,117)( 6,116)( 7,112)
( 8,114)( 9,113)( 10,118)( 11,120)( 12,119)( 13,124)( 14,126)( 15,125)
( 16,121)( 17,123)( 18,122)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)
( 24,134)( 25,130)( 26,132)( 27,131)( 28,136)( 29,138)( 30,137)( 31,142)
( 32,144)( 33,143)( 34,139)( 35,141)( 36,140)( 37,145)( 38,147)( 39,146)
( 40,151)( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,154)( 47,156)
( 48,155)( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,163)
( 56,165)( 57,164)( 58,169)( 59,171)( 60,170)( 61,166)( 62,168)( 63,167)
( 64,172)( 65,174)( 66,173)( 67,178)( 68,180)( 69,179)( 70,175)( 71,177)
( 72,176)( 73,181)( 74,183)( 75,182)( 76,187)( 77,189)( 78,188)( 79,184)
( 80,186)( 81,185)( 82,190)( 83,192)( 84,191)( 85,196)( 86,198)( 87,197)
( 88,193)( 89,195)( 90,194)( 91,199)( 92,201)( 93,200)( 94,205)( 95,207)
( 96,206)( 97,202)( 98,204)( 99,203)(100,208)(101,210)(102,209)(103,214)
(104,216)(105,215)(106,211)(107,213)(108,212);
s4 := Sym(218)!(217,218);
poly := sub<Sym(218)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
to this polytope