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# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*432g
if this polytope has a name.
Group : SmallGroup(432,602)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 36, 108, 18
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,6,2} of size 864
{12,6,3} of size 1296
{12,6,4} of size 1728
{12,6,4} of size 1728
Vertex Figure Of :
{2,12,6} of size 864
{4,12,6} of size 1728
{4,12,6} of size 1728
{4,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*216d
3-fold quotients : {12,6}*144a, {12,6}*144b, {12,6}*144c
6-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
9-fold quotients : {12,2}*48, {4,6}*48a
12-fold quotients : {3,6}*36, {6,3}*36
18-fold quotients : {2,6}*24, {6,2}*24
27-fold quotients : {4,2}*16
36-fold quotients : {2,3}*12, {3,2}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,6}*864f, {12,12}*864h
3-fold covers : {36,6}*1296l, {12,18}*1296l, {12,6}*1296g, {12,6}*1296h, {12,6}*1296i
4-fold covers : {48,6}*1728f, {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h, {12,12}*1728v, {12,6}*1728h, {12,6}*1728i
Permutation Representation (GAP) :
```s0 := (  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)( 55,190)( 56,192)
( 57,191)( 58,196)( 59,198)( 60,197)( 61,193)( 62,195)( 63,194)( 64,199)
( 65,201)( 66,200)( 67,205)( 68,207)( 69,206)( 70,202)( 71,204)( 72,203)
( 73,208)( 74,210)( 75,209)( 76,214)( 77,216)( 78,215)( 79,211)( 80,213)
( 81,212)( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)
( 89,168)( 90,167)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)
( 97,175)( 98,177)( 99,176)(100,181)(101,183)(102,182)(103,187)(104,189)
(105,188)(106,184)(107,186)(108,185);;
s1 := (  1,167)(  2,166)(  3,168)(  4,164)(  5,163)(  6,165)(  7,170)(  8,169)
(  9,171)( 10,185)( 11,184)( 12,186)( 13,182)( 14,181)( 15,183)( 16,188)
( 17,187)( 18,189)( 19,176)( 20,175)( 21,177)( 22,173)( 23,172)( 24,174)
( 25,179)( 26,178)( 27,180)( 28,194)( 29,193)( 30,195)( 31,191)( 32,190)
( 33,192)( 34,197)( 35,196)( 36,198)( 37,212)( 38,211)( 39,213)( 40,209)
( 41,208)( 42,210)( 43,215)( 44,214)( 45,216)( 46,203)( 47,202)( 48,204)
( 49,200)( 50,199)( 51,201)( 52,206)( 53,205)( 54,207)( 55,113)( 56,112)
( 57,114)( 58,110)( 59,109)( 60,111)( 61,116)( 62,115)( 63,117)( 64,131)
( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,134)( 71,133)( 72,135)
( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,125)( 80,124)
( 81,126)( 82,140)( 83,139)( 84,141)( 85,137)( 86,136)( 87,138)( 88,143)
( 89,142)( 90,144)( 91,158)( 92,157)( 93,159)( 94,155)( 95,154)( 96,156)
( 97,161)( 98,160)( 99,162)(100,149)(101,148)(102,150)(103,146)(104,145)
(105,147)(106,152)(107,151)(108,153);;
s2 := (  1,172)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)(  8,176)
(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)( 16,166)
( 17,167)( 18,168)( 19,181)( 20,182)( 21,183)( 22,187)( 23,188)( 24,189)
( 25,184)( 26,185)( 27,186)( 28,199)( 29,200)( 30,201)( 31,205)( 32,206)
( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)( 40,196)
( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)( 48,210)
( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,145)( 56,146)
( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)( 64,136)
( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)( 72,141)
( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)
( 81,159)( 82,118)( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)( 88,121)
( 89,122)( 90,123)( 91,109)( 92,110)( 93,111)( 94,115)( 95,116)( 96,117)
( 97,112)( 98,113)( 99,114)(100,127)(101,128)(102,129)(103,133)(104,134)
(105,135)(106,130)(107,131)(108,132);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
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(216)!(  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)( 55,190)
( 56,192)( 57,191)( 58,196)( 59,198)( 60,197)( 61,193)( 62,195)( 63,194)
( 64,199)( 65,201)( 66,200)( 67,205)( 68,207)( 69,206)( 70,202)( 71,204)
( 72,203)( 73,208)( 74,210)( 75,209)( 76,214)( 77,216)( 78,215)( 79,211)
( 80,213)( 81,212)( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)
( 88,166)( 89,168)( 90,167)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)
( 96,179)( 97,175)( 98,177)( 99,176)(100,181)(101,183)(102,182)(103,187)
(104,189)(105,188)(106,184)(107,186)(108,185);
s1 := Sym(216)!(  1,167)(  2,166)(  3,168)(  4,164)(  5,163)(  6,165)(  7,170)
(  8,169)(  9,171)( 10,185)( 11,184)( 12,186)( 13,182)( 14,181)( 15,183)
( 16,188)( 17,187)( 18,189)( 19,176)( 20,175)( 21,177)( 22,173)( 23,172)
( 24,174)( 25,179)( 26,178)( 27,180)( 28,194)( 29,193)( 30,195)( 31,191)
( 32,190)( 33,192)( 34,197)( 35,196)( 36,198)( 37,212)( 38,211)( 39,213)
( 40,209)( 41,208)( 42,210)( 43,215)( 44,214)( 45,216)( 46,203)( 47,202)
( 48,204)( 49,200)( 50,199)( 51,201)( 52,206)( 53,205)( 54,207)( 55,113)
( 56,112)( 57,114)( 58,110)( 59,109)( 60,111)( 61,116)( 62,115)( 63,117)
( 64,131)( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,134)( 71,133)
( 72,135)( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,125)
( 80,124)( 81,126)( 82,140)( 83,139)( 84,141)( 85,137)( 86,136)( 87,138)
( 88,143)( 89,142)( 90,144)( 91,158)( 92,157)( 93,159)( 94,155)( 95,154)
( 96,156)( 97,161)( 98,160)( 99,162)(100,149)(101,148)(102,150)(103,146)
(104,145)(105,147)(106,152)(107,151)(108,153);
s2 := Sym(216)!(  1,172)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)
(  8,176)(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)
( 16,166)( 17,167)( 18,168)( 19,181)( 20,182)( 21,183)( 22,187)( 23,188)
( 24,189)( 25,184)( 26,185)( 27,186)( 28,199)( 29,200)( 30,201)( 31,205)
( 32,206)( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)
( 40,196)( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)
( 48,210)( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,145)
( 56,146)( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)
( 64,136)( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)
( 72,141)( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)
( 80,158)( 81,159)( 82,118)( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)
( 88,121)( 89,122)( 90,123)( 91,109)( 92,110)( 93,111)( 94,115)( 95,116)
( 96,117)( 97,112)( 98,113)( 99,114)(100,127)(101,128)(102,129)(103,133)
(104,134)(105,135)(106,130)(107,131)(108,132);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope