Polytope of Type {2,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,6}*1728b
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 : {2,12,6}*864d
   3-fold quotients : {2,12,6}*576a
   4-fold quotients : {2,6,6}*432b
   6-fold quotients : {2,12,6}*288d
   8-fold quotients : {2,6,6}*216
   9-fold quotients : {2,4,6}*192
   12-fold quotients : {2,6,6}*144a
   18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   36-fold quotients : {2,4,3}*48, {2,2,6}*48, {2,6,2}*48
   72-fold quotients : {2,2,3}*24, {2,3,2}*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)(  7,191)(  8,194)(  9,193)( 10,192)
( 11,187)( 12,190)( 13,189)( 14,188)( 15,203)( 16,206)( 17,205)( 18,204)
( 19,199)( 20,202)( 21,201)( 22,200)( 23,195)( 24,198)( 25,197)( 26,196)
( 27,211)( 28,214)( 29,213)( 30,212)( 31,207)( 32,210)( 33,209)( 34,208)
( 35,215)( 36,218)( 37,217)( 38,216)( 39,147)( 40,150)( 41,149)( 42,148)
( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)( 50,152)
( 51,167)( 52,170)( 53,169)( 54,168)( 55,163)( 56,166)( 57,165)( 58,164)
( 59,159)( 60,162)( 61,161)( 62,160)( 63,175)( 64,178)( 65,177)( 66,176)
( 67,171)( 68,174)( 69,173)( 70,172)( 71,179)( 72,182)( 73,181)( 74,180)
( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)( 82,120)
( 83,115)( 84,118)( 85,117)( 86,116)( 87,131)( 88,134)( 89,133)( 90,132)
( 91,127)( 92,130)( 93,129)( 94,128)( 95,123)( 96,126)( 97,125)( 98,124)
( 99,139)(100,142)(101,141)(102,140)(103,135)(104,138)(105,137)(106,136)
(107,143)(108,146)(109,145)(110,144);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*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)(  7,191)(  8,194)(  9,193)
( 10,192)( 11,187)( 12,190)( 13,189)( 14,188)( 15,203)( 16,206)( 17,205)
( 18,204)( 19,199)( 20,202)( 21,201)( 22,200)( 23,195)( 24,198)( 25,197)
( 26,196)( 27,211)( 28,214)( 29,213)( 30,212)( 31,207)( 32,210)( 33,209)
( 34,208)( 35,215)( 36,218)( 37,217)( 38,216)( 39,147)( 40,150)( 41,149)
( 42,148)( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)
( 50,152)( 51,167)( 52,170)( 53,169)( 54,168)( 55,163)( 56,166)( 57,165)
( 58,164)( 59,159)( 60,162)( 61,161)( 62,160)( 63,175)( 64,178)( 65,177)
( 66,176)( 67,171)( 68,174)( 69,173)( 70,172)( 71,179)( 72,182)( 73,181)
( 74,180)( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)
( 82,120)( 83,115)( 84,118)( 85,117)( 86,116)( 87,131)( 88,134)( 89,133)
( 90,132)( 91,127)( 92,130)( 93,129)( 94,128)( 95,123)( 96,126)( 97,125)
( 98,124)( 99,139)(100,142)(101,141)(102,140)(103,135)(104,138)(105,137)
(106,136)(107,143)(108,146)(109,145)(110,144);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*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