Polytope of Type {2,6,12,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,12,3}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 5
Schlafli Type : {2,6,12,3}
Number of vertices, edges, etc : 2, 6, 72, 36, 6
Order of s0s1s2s3s4 : 6
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) :
3-fold quotients : {2,6,4,3}*576
4-fold quotients : {2,6,6,3}*432a
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,2,4,3}*192
12-fold quotients : {2,6,2,3}*144
18-fold quotients : {2,2,4,3}*96
24-fold quotients : {2,3,2,3}*72
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 27)( 16, 28)( 17, 29)( 18, 30)
( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 31)( 24, 32)( 25, 33)( 26, 34)
( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 71)( 56, 72)( 57, 73)( 58, 74)( 59, 67)( 60, 68)( 61, 69)( 62, 70)
( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87, 99)( 88,100)( 89,101)( 90,102)
( 91,107)( 92,108)( 93,109)( 94,110)( 95,103)( 96,104)( 97,105)( 98,106)
(115,119)(116,120)(117,121)(118,122)(123,135)(124,136)(125,137)(126,138)
(127,143)(128,144)(129,145)(130,146)(131,139)(132,140)(133,141)(134,142)
(151,155)(152,156)(153,157)(154,158)(159,171)(160,172)(161,173)(162,174)
(163,179)(164,180)(165,181)(166,182)(167,175)(168,176)(169,177)(170,178)
(187,191)(188,192)(189,193)(190,194)(195,207)(196,208)(197,209)(198,210)
(199,215)(200,216)(201,217)(202,218)(203,211)(204,212)(205,213)(206,214);;
s2 := ( 3,129)( 4,130)( 5,127)( 6,128)( 7,125)( 8,126)( 9,123)( 10,124)
( 11,133)( 12,134)( 13,131)( 14,132)( 15,117)( 16,118)( 17,115)( 18,116)
( 19,113)( 20,114)( 21,111)( 22,112)( 23,121)( 24,122)( 25,119)( 26,120)
( 27,141)( 28,142)( 29,139)( 30,140)( 31,137)( 32,138)( 33,135)( 34,136)
( 35,145)( 36,146)( 37,143)( 38,144)( 39,165)( 40,166)( 41,163)( 42,164)
( 43,161)( 44,162)( 45,159)( 46,160)( 47,169)( 48,170)( 49,167)( 50,168)
( 51,153)( 52,154)( 53,151)( 54,152)( 55,149)( 56,150)( 57,147)( 58,148)
( 59,157)( 60,158)( 61,155)( 62,156)( 63,177)( 64,178)( 65,175)( 66,176)
( 67,173)( 68,174)( 69,171)( 70,172)( 71,181)( 72,182)( 73,179)( 74,180)
( 75,201)( 76,202)( 77,199)( 78,200)( 79,197)( 80,198)( 81,195)( 82,196)
( 83,205)( 84,206)( 85,203)( 86,204)( 87,189)( 88,190)( 89,187)( 90,188)
( 91,185)( 92,186)( 93,183)( 94,184)( 95,193)( 96,194)( 97,191)( 98,192)
( 99,213)(100,214)(101,211)(102,212)(103,209)(104,210)(105,207)(106,208)
(107,217)(108,218)(109,215)(110,216);;
s3 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)( 47, 79)
( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)
( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)
( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)(118,122)
(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)
(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)(152,193)
(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)(160,197)
(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)(168,201)
(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)(176,217)
(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);;
s4 := ( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 83)( 8, 86)( 9, 85)( 10, 84)
( 11, 79)( 12, 82)( 13, 81)( 14, 80)( 15, 95)( 16, 98)( 17, 97)( 18, 96)
( 19, 91)( 20, 94)( 21, 93)( 22, 92)( 23, 87)( 24, 90)( 25, 89)( 26, 88)
( 27,103)( 28,106)( 29,105)( 30,104)( 31, 99)( 32,102)( 33,101)( 34,100)
( 35,107)( 36,110)( 37,109)( 38,108)( 40, 42)( 43, 47)( 44, 50)( 45, 49)
( 46, 48)( 51, 59)( 52, 62)( 53, 61)( 54, 60)( 56, 58)( 63, 67)( 64, 70)
( 65, 69)( 66, 68)( 72, 74)(111,183)(112,186)(113,185)(114,184)(115,191)
(116,194)(117,193)(118,192)(119,187)(120,190)(121,189)(122,188)(123,203)
(124,206)(125,205)(126,204)(127,199)(128,202)(129,201)(130,200)(131,195)
(132,198)(133,197)(134,196)(135,211)(136,214)(137,213)(138,212)(139,207)
(140,210)(141,209)(142,208)(143,215)(144,218)(145,217)(146,216)(148,150)
(151,155)(152,158)(153,157)(154,156)(159,167)(160,170)(161,169)(162,168)
(164,166)(171,175)(172,178)(173,177)(174,176)(180,182);;
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*s1*s0*s1,
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*s3*s4, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s4*s3*s2*s1*s2*s3*s1*s4*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 27)( 16, 28)( 17, 29)
( 18, 30)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 31)( 24, 32)( 25, 33)
( 26, 34)( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 71)( 56, 72)( 57, 73)( 58, 74)( 59, 67)( 60, 68)( 61, 69)
( 62, 70)( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87, 99)( 88,100)( 89,101)
( 90,102)( 91,107)( 92,108)( 93,109)( 94,110)( 95,103)( 96,104)( 97,105)
( 98,106)(115,119)(116,120)(117,121)(118,122)(123,135)(124,136)(125,137)
(126,138)(127,143)(128,144)(129,145)(130,146)(131,139)(132,140)(133,141)
(134,142)(151,155)(152,156)(153,157)(154,158)(159,171)(160,172)(161,173)
(162,174)(163,179)(164,180)(165,181)(166,182)(167,175)(168,176)(169,177)
(170,178)(187,191)(188,192)(189,193)(190,194)(195,207)(196,208)(197,209)
(198,210)(199,215)(200,216)(201,217)(202,218)(203,211)(204,212)(205,213)
(206,214);
s2 := Sym(218)!( 3,129)( 4,130)( 5,127)( 6,128)( 7,125)( 8,126)( 9,123)
( 10,124)( 11,133)( 12,134)( 13,131)( 14,132)( 15,117)( 16,118)( 17,115)
( 18,116)( 19,113)( 20,114)( 21,111)( 22,112)( 23,121)( 24,122)( 25,119)
( 26,120)( 27,141)( 28,142)( 29,139)( 30,140)( 31,137)( 32,138)( 33,135)
( 34,136)( 35,145)( 36,146)( 37,143)( 38,144)( 39,165)( 40,166)( 41,163)
( 42,164)( 43,161)( 44,162)( 45,159)( 46,160)( 47,169)( 48,170)( 49,167)
( 50,168)( 51,153)( 52,154)( 53,151)( 54,152)( 55,149)( 56,150)( 57,147)
( 58,148)( 59,157)( 60,158)( 61,155)( 62,156)( 63,177)( 64,178)( 65,175)
( 66,176)( 67,173)( 68,174)( 69,171)( 70,172)( 71,181)( 72,182)( 73,179)
( 74,180)( 75,201)( 76,202)( 77,199)( 78,200)( 79,197)( 80,198)( 81,195)
( 82,196)( 83,205)( 84,206)( 85,203)( 86,204)( 87,189)( 88,190)( 89,187)
( 90,188)( 91,185)( 92,186)( 93,183)( 94,184)( 95,193)( 96,194)( 97,191)
( 98,192)( 99,213)(100,214)(101,211)(102,212)(103,209)(104,210)(105,207)
(106,208)(107,217)(108,218)(109,215)(110,216);
s3 := Sym(218)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)
( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)
( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)
( 71,103)( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)
(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)
(140,145)(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)
(152,193)(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)
(160,197)(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)
(168,201)(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)
(176,217)(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);
s4 := Sym(218)!( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 83)( 8, 86)( 9, 85)
( 10, 84)( 11, 79)( 12, 82)( 13, 81)( 14, 80)( 15, 95)( 16, 98)( 17, 97)
( 18, 96)( 19, 91)( 20, 94)( 21, 93)( 22, 92)( 23, 87)( 24, 90)( 25, 89)
( 26, 88)( 27,103)( 28,106)( 29,105)( 30,104)( 31, 99)( 32,102)( 33,101)
( 34,100)( 35,107)( 36,110)( 37,109)( 38,108)( 40, 42)( 43, 47)( 44, 50)
( 45, 49)( 46, 48)( 51, 59)( 52, 62)( 53, 61)( 54, 60)( 56, 58)( 63, 67)
( 64, 70)( 65, 69)( 66, 68)( 72, 74)(111,183)(112,186)(113,185)(114,184)
(115,191)(116,194)(117,193)(118,192)(119,187)(120,190)(121,189)(122,188)
(123,203)(124,206)(125,205)(126,204)(127,199)(128,202)(129,201)(130,200)
(131,195)(132,198)(133,197)(134,196)(135,211)(136,214)(137,213)(138,212)
(139,207)(140,210)(141,209)(142,208)(143,215)(144,218)(145,217)(146,216)
(148,150)(151,155)(152,158)(153,157)(154,156)(159,167)(160,170)(161,169)
(162,168)(164,166)(171,175)(172,178)(173,177)(174,176)(180,182);
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*s1*s0*s1, 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*s3*s4,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s4*s3*s2*s1*s2*s3*s1*s4*s2*s1*s2*s3*s1*s2*s1*s2 >;
to this polytope