Polytope of Type {2,12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 4
Schlafli Type : {2,12,6}
Number of vertices, edges, etc : 2, 72, 216, 36
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 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 : {2,12,3}*864
3-fold quotients : {2,12,6}*576b
4-fold quotients : {2,6,6}*432a
6-fold quotients : {2,12,3}*288
8-fold quotients : {2,6,3}*216
9-fold quotients : {2,4,6}*192
12-fold quotients : {2,6,6}*144b
18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
24-fold quotients : {2,6,3}*72
36-fold quotients : {2,4,3}*48, {2,2,6}*48
72-fold quotients : {2,2,3}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 5)( 4, 6)( 7, 13)( 8, 14)( 9, 11)( 10, 12)( 15, 29)( 16, 30)
( 17, 27)( 18, 28)( 19, 37)( 20, 38)( 21, 35)( 22, 36)( 23, 33)( 24, 34)
( 25, 31)( 26, 32)( 39, 41)( 40, 42)( 43, 49)( 44, 50)( 45, 47)( 46, 48)
( 51, 65)( 52, 66)( 53, 63)( 54, 64)( 55, 73)( 56, 74)( 57, 71)( 58, 72)
( 59, 69)( 60, 70)( 61, 67)( 62, 68)( 75, 77)( 76, 78)( 79, 85)( 80, 86)
( 81, 83)( 82, 84)( 87,101)( 88,102)( 89, 99)( 90,100)( 91,109)( 92,110)
( 93,107)( 94,108)( 95,105)( 96,106)( 97,103)( 98,104)(111,113)(112,114)
(115,121)(116,122)(117,119)(118,120)(123,137)(124,138)(125,135)(126,136)
(127,145)(128,146)(129,143)(130,144)(131,141)(132,142)(133,139)(134,140)
(147,149)(148,150)(151,157)(152,158)(153,155)(154,156)(159,173)(160,174)
(161,171)(162,172)(163,181)(164,182)(165,179)(166,180)(167,177)(168,178)
(169,175)(170,176)(183,185)(184,186)(187,193)(188,194)(189,191)(190,192)
(195,209)(196,210)(197,207)(198,208)(199,217)(200,218)(201,215)(202,216)
(203,213)(204,214)(205,211)(206,212);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 19)( 8, 21)( 9, 20)( 10, 22)
( 11, 23)( 12, 25)( 13, 24)( 14, 26)( 28, 29)( 32, 33)( 36, 37)( 39, 87)
( 40, 89)( 41, 88)( 42, 90)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 95)
( 48, 97)( 49, 96)( 50, 98)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 79)
( 56, 81)( 57, 80)( 58, 82)( 59, 83)( 60, 85)( 61, 84)( 62, 86)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)( 70,106)( 71,107)
( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)(114,126)(115,127)
(116,129)(117,128)(118,130)(119,131)(120,133)(121,132)(122,134)(136,137)
(140,141)(144,145)(147,195)(148,197)(149,196)(150,198)(151,199)(152,201)
(153,200)(154,202)(155,203)(156,205)(157,204)(158,206)(159,183)(160,185)
(161,184)(162,186)(163,187)(164,189)(165,188)(166,190)(167,191)(168,193)
(169,192)(170,194)(171,207)(172,209)(173,208)(174,210)(175,211)(176,213)
(177,212)(178,214)(179,215)(180,217)(181,216)(182,218);;
s3 := ( 3,183)( 4,186)( 5,185)( 6,184)( 7,187)( 8,190)( 9,189)( 10,188)
( 11,191)( 12,194)( 13,193)( 14,192)( 15,211)( 16,214)( 17,213)( 18,212)
( 19,215)( 20,218)( 21,217)( 22,216)( 23,207)( 24,210)( 25,209)( 26,208)
( 27,203)( 28,206)( 29,205)( 30,204)( 31,195)( 32,198)( 33,197)( 34,196)
( 35,199)( 36,202)( 37,201)( 38,200)( 39,147)( 40,150)( 41,149)( 42,148)
( 43,151)( 44,154)( 45,153)( 46,152)( 47,155)( 48,158)( 49,157)( 50,156)
( 51,175)( 52,178)( 53,177)( 54,176)( 55,179)( 56,182)( 57,181)( 58,180)
( 59,171)( 60,174)( 61,173)( 62,172)( 63,167)( 64,170)( 65,169)( 66,168)
( 67,159)( 68,162)( 69,161)( 70,160)( 71,163)( 72,166)( 73,165)( 74,164)
( 75,111)( 76,114)( 77,113)( 78,112)( 79,115)( 80,118)( 81,117)( 82,116)
( 83,119)( 84,122)( 85,121)( 86,120)( 87,139)( 88,142)( 89,141)( 90,140)
( 91,143)( 92,146)( 93,145)( 94,144)( 95,135)( 96,138)( 97,137)( 98,136)
( 99,131)(100,134)(101,133)(102,132)(103,123)(104,126)(105,125)(106,124)
(107,127)(108,130)(109,129)(110,128);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 3, 5)( 4, 6)( 7, 13)( 8, 14)( 9, 11)( 10, 12)( 15, 29)
( 16, 30)( 17, 27)( 18, 28)( 19, 37)( 20, 38)( 21, 35)( 22, 36)( 23, 33)
( 24, 34)( 25, 31)( 26, 32)( 39, 41)( 40, 42)( 43, 49)( 44, 50)( 45, 47)
( 46, 48)( 51, 65)( 52, 66)( 53, 63)( 54, 64)( 55, 73)( 56, 74)( 57, 71)
( 58, 72)( 59, 69)( 60, 70)( 61, 67)( 62, 68)( 75, 77)( 76, 78)( 79, 85)
( 80, 86)( 81, 83)( 82, 84)( 87,101)( 88,102)( 89, 99)( 90,100)( 91,109)
( 92,110)( 93,107)( 94,108)( 95,105)( 96,106)( 97,103)( 98,104)(111,113)
(112,114)(115,121)(116,122)(117,119)(118,120)(123,137)(124,138)(125,135)
(126,136)(127,145)(128,146)(129,143)(130,144)(131,141)(132,142)(133,139)
(134,140)(147,149)(148,150)(151,157)(152,158)(153,155)(154,156)(159,173)
(160,174)(161,171)(162,172)(163,181)(164,182)(165,179)(166,180)(167,177)
(168,178)(169,175)(170,176)(183,185)(184,186)(187,193)(188,194)(189,191)
(190,192)(195,209)(196,210)(197,207)(198,208)(199,217)(200,218)(201,215)
(202,216)(203,213)(204,214)(205,211)(206,212);
s2 := Sym(218)!( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 19)( 8, 21)( 9, 20)
( 10, 22)( 11, 23)( 12, 25)( 13, 24)( 14, 26)( 28, 29)( 32, 33)( 36, 37)
( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 91)( 44, 93)( 45, 92)( 46, 94)
( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 75)( 52, 77)( 53, 76)( 54, 78)
( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 83)( 60, 85)( 61, 84)( 62, 86)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)(114,126)
(115,127)(116,129)(117,128)(118,130)(119,131)(120,133)(121,132)(122,134)
(136,137)(140,141)(144,145)(147,195)(148,197)(149,196)(150,198)(151,199)
(152,201)(153,200)(154,202)(155,203)(156,205)(157,204)(158,206)(159,183)
(160,185)(161,184)(162,186)(163,187)(164,189)(165,188)(166,190)(167,191)
(168,193)(169,192)(170,194)(171,207)(172,209)(173,208)(174,210)(175,211)
(176,213)(177,212)(178,214)(179,215)(180,217)(181,216)(182,218);
s3 := Sym(218)!( 3,183)( 4,186)( 5,185)( 6,184)( 7,187)( 8,190)( 9,189)
( 10,188)( 11,191)( 12,194)( 13,193)( 14,192)( 15,211)( 16,214)( 17,213)
( 18,212)( 19,215)( 20,218)( 21,217)( 22,216)( 23,207)( 24,210)( 25,209)
( 26,208)( 27,203)( 28,206)( 29,205)( 30,204)( 31,195)( 32,198)( 33,197)
( 34,196)( 35,199)( 36,202)( 37,201)( 38,200)( 39,147)( 40,150)( 41,149)
( 42,148)( 43,151)( 44,154)( 45,153)( 46,152)( 47,155)( 48,158)( 49,157)
( 50,156)( 51,175)( 52,178)( 53,177)( 54,176)( 55,179)( 56,182)( 57,181)
( 58,180)( 59,171)( 60,174)( 61,173)( 62,172)( 63,167)( 64,170)( 65,169)
( 66,168)( 67,159)( 68,162)( 69,161)( 70,160)( 71,163)( 72,166)( 73,165)
( 74,164)( 75,111)( 76,114)( 77,113)( 78,112)( 79,115)( 80,118)( 81,117)
( 82,116)( 83,119)( 84,122)( 85,121)( 86,120)( 87,139)( 88,142)( 89,141)
( 90,140)( 91,143)( 92,146)( 93,145)( 94,144)( 95,135)( 96,138)( 97,137)
( 98,136)( 99,131)(100,134)(101,133)(102,132)(103,123)(104,126)(105,125)
(106,124)(107,127)(108,130)(109,129)(110,128);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, 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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope