Polytope of Type {6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*432g
if this polytope has a name.
Group : SmallGroup(432,602)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 18, 108, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,2} of size 864
{6,12,4} of size 1728
{6,12,4} of size 1728
{6,12,4} of size 1728
Vertex Figure Of :
{2,6,12} of size 864
{3,6,12} of size 1296
{4,6,12} of size 1728
{4,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*216d
3-fold quotients : {6,12}*144a, {6,12}*144b, {6,12}*144c
6-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
9-fold quotients : {2,12}*48, {6,4}*48a
12-fold quotients : {3,6}*36, {6,3}*36
18-fold quotients : {2,6}*24, {6,2}*24
27-fold quotients : {2,4}*16
36-fold quotients : {2,3}*12, {3,2}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,24}*864f, {12,12}*864h
3-fold covers : {6,36}*1296l, {18,12}*1296l, {6,12}*1296g, {6,12}*1296h, {6,12}*1296i
4-fold covers : {6,48}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {12,12}*1728w, {6,12}*1728h, {6,12}*1728i
Permutation Representation (GAP) :
s0 := ( 1,163)( 2,164)( 3,165)( 4,169)( 5,170)( 6,171)( 7,166)( 8,167)
( 9,168)( 10,181)( 11,182)( 12,183)( 13,187)( 14,188)( 15,189)( 16,184)
( 17,185)( 18,186)( 19,172)( 20,173)( 21,174)( 22,178)( 23,179)( 24,180)
( 25,175)( 26,176)( 27,177)( 28,190)( 29,191)( 30,192)( 31,196)( 32,197)
( 33,198)( 34,193)( 35,194)( 36,195)( 37,208)( 38,209)( 39,210)( 40,214)
( 41,215)( 42,216)( 43,211)( 44,212)( 45,213)( 46,199)( 47,200)( 48,201)
( 49,205)( 50,206)( 51,207)( 52,202)( 53,203)( 54,204)( 55,136)( 56,137)
( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)( 64,154)
( 65,155)( 66,156)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)( 72,159)
( 73,145)( 74,146)( 75,147)( 76,151)( 77,152)( 78,153)( 79,148)( 80,149)
( 81,150)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)
( 89,113)( 90,114)( 91,127)( 92,128)( 93,129)( 94,133)( 95,134)( 96,135)
( 97,130)( 98,131)( 99,132)(100,118)(101,119)(102,120)(103,124)(104,125)
(105,126)(106,121)(107,122)(108,123);;
s1 := ( 1,175)( 2,177)( 3,176)( 4,172)( 5,174)( 6,173)( 7,178)( 8,180)
( 9,179)( 10,166)( 11,168)( 12,167)( 13,163)( 14,165)( 15,164)( 16,169)
( 17,171)( 18,170)( 19,184)( 20,186)( 21,185)( 22,181)( 23,183)( 24,182)
( 25,187)( 26,189)( 27,188)( 28,202)( 29,204)( 30,203)( 31,199)( 32,201)
( 33,200)( 34,205)( 35,207)( 36,206)( 37,193)( 38,195)( 39,194)( 40,190)
( 41,192)( 42,191)( 43,196)( 44,198)( 45,197)( 46,211)( 47,213)( 48,212)
( 49,208)( 50,210)( 51,209)( 52,214)( 53,216)( 54,215)( 55,121)( 56,123)
( 57,122)( 58,118)( 59,120)( 60,119)( 61,124)( 62,126)( 63,125)( 64,112)
( 65,114)( 66,113)( 67,109)( 68,111)( 69,110)( 70,115)( 71,117)( 72,116)
( 73,130)( 74,132)( 75,131)( 76,127)( 77,129)( 78,128)( 79,133)( 80,135)
( 81,134)( 82,148)( 83,150)( 84,149)( 85,145)( 86,147)( 87,146)( 88,151)
( 89,153)( 90,152)( 91,139)( 92,141)( 93,140)( 94,136)( 95,138)( 96,137)
( 97,142)( 98,144)( 99,143)(100,157)(101,159)(102,158)(103,154)(104,156)
(105,155)(106,160)(107,162)(108,161);;
s2 := ( 1,110)( 2,109)( 3,111)( 4,116)( 5,115)( 6,117)( 7,113)( 8,112)
( 9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)
( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)( 24,135)
( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)
( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)( 40,152)
( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)( 48,156)
( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,191)( 56,190)
( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)( 64,200)
( 65,199)( 66,201)( 67,206)( 68,205)( 69,207)( 70,203)( 71,202)( 72,204)
( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)( 80,211)
( 81,213)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)
( 89,166)( 90,168)( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)( 96,180)
( 97,176)( 98,175)( 99,177)(100,182)(101,181)(102,183)(103,188)(104,187)
(105,189)(106,185)(107,184)(108,186);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
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(216)!( 1,163)( 2,164)( 3,165)( 4,169)( 5,170)( 6,171)( 7,166)
( 8,167)( 9,168)( 10,181)( 11,182)( 12,183)( 13,187)( 14,188)( 15,189)
( 16,184)( 17,185)( 18,186)( 19,172)( 20,173)( 21,174)( 22,178)( 23,179)
( 24,180)( 25,175)( 26,176)( 27,177)( 28,190)( 29,191)( 30,192)( 31,196)
( 32,197)( 33,198)( 34,193)( 35,194)( 36,195)( 37,208)( 38,209)( 39,210)
( 40,214)( 41,215)( 42,216)( 43,211)( 44,212)( 45,213)( 46,199)( 47,200)
( 48,201)( 49,205)( 50,206)( 51,207)( 52,202)( 53,203)( 54,204)( 55,136)
( 56,137)( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)
( 64,154)( 65,155)( 66,156)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)
( 72,159)( 73,145)( 74,146)( 75,147)( 76,151)( 77,152)( 78,153)( 79,148)
( 80,149)( 81,150)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)
( 88,112)( 89,113)( 90,114)( 91,127)( 92,128)( 93,129)( 94,133)( 95,134)
( 96,135)( 97,130)( 98,131)( 99,132)(100,118)(101,119)(102,120)(103,124)
(104,125)(105,126)(106,121)(107,122)(108,123);
s1 := Sym(216)!( 1,175)( 2,177)( 3,176)( 4,172)( 5,174)( 6,173)( 7,178)
( 8,180)( 9,179)( 10,166)( 11,168)( 12,167)( 13,163)( 14,165)( 15,164)
( 16,169)( 17,171)( 18,170)( 19,184)( 20,186)( 21,185)( 22,181)( 23,183)
( 24,182)( 25,187)( 26,189)( 27,188)( 28,202)( 29,204)( 30,203)( 31,199)
( 32,201)( 33,200)( 34,205)( 35,207)( 36,206)( 37,193)( 38,195)( 39,194)
( 40,190)( 41,192)( 42,191)( 43,196)( 44,198)( 45,197)( 46,211)( 47,213)
( 48,212)( 49,208)( 50,210)( 51,209)( 52,214)( 53,216)( 54,215)( 55,121)
( 56,123)( 57,122)( 58,118)( 59,120)( 60,119)( 61,124)( 62,126)( 63,125)
( 64,112)( 65,114)( 66,113)( 67,109)( 68,111)( 69,110)( 70,115)( 71,117)
( 72,116)( 73,130)( 74,132)( 75,131)( 76,127)( 77,129)( 78,128)( 79,133)
( 80,135)( 81,134)( 82,148)( 83,150)( 84,149)( 85,145)( 86,147)( 87,146)
( 88,151)( 89,153)( 90,152)( 91,139)( 92,141)( 93,140)( 94,136)( 95,138)
( 96,137)( 97,142)( 98,144)( 99,143)(100,157)(101,159)(102,158)(103,154)
(104,156)(105,155)(106,160)(107,162)(108,161);
s2 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,116)( 5,115)( 6,117)( 7,113)
( 8,112)( 9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)
( 16,122)( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)
( 24,135)( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)
( 32,142)( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)
( 40,152)( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)
( 48,156)( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,191)
( 56,190)( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)
( 64,200)( 65,199)( 66,201)( 67,206)( 68,205)( 69,207)( 70,203)( 71,202)
( 72,204)( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)
( 80,211)( 81,213)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)
( 88,167)( 89,166)( 90,168)( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)
( 96,180)( 97,176)( 98,175)( 99,177)(100,182)(101,181)(102,183)(103,188)
(104,187)(105,189)(106,185)(107,184)(108,186);
poly := sub<Sym(216)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope