# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*576e
if this polytope has a name.
Group : SmallGroup(576,8355)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 48, 144, 24
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,6,2} of size 1152
Vertex Figure Of :
{2,12,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*288a
3-fold quotients : {12,6}*192a
4-fold quotients : {12,6}*144c, {6,3}*144
6-fold quotients : {6,6}*96
8-fold quotients : {6,6}*72b
12-fold quotients : {4,6}*48a, {3,6}*48, {6,3}*48
16-fold quotients : {6,3}*36
24-fold quotients : {3,3}*24, {2,6}*24
36-fold quotients : {4,2}*16
48-fold quotients : {2,3}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*1152j, {24,6}*1152j, {12,6}*1152e, {12,12}*1152p, {24,6}*1152m
3-fold covers : {12,18}*1728b, {12,6}*1728e, {12,6}*1728h
Permutation Representation (GAP) :
```s0 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 38, 39)
( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 73,109)( 74,111)
( 75,110)( 76,112)( 77,113)( 78,115)( 79,114)( 80,116)( 81,117)( 82,119)
( 83,118)( 84,120)( 85,133)( 86,135)( 87,134)( 88,136)( 89,137)( 90,139)
( 91,138)( 92,140)( 93,141)( 94,143)( 95,142)( 96,144)( 97,121)( 98,123)
( 99,122)(100,124)(101,125)(102,127)(103,126)(104,128)(105,129)(106,131)
(107,130)(108,132);;
s1 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 93)(  6, 94)(  7, 96)(  8, 95)
(  9, 89)( 10, 90)( 11, 92)( 12, 91)( 13, 73)( 14, 74)( 15, 76)( 16, 75)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)( 24, 79)
( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)( 32,107)
( 33,101)( 34,102)( 35,104)( 36,103)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)( 56,119)
( 57,113)( 58,114)( 59,116)( 60,115)( 61,133)( 62,134)( 63,136)( 64,135)
( 65,141)( 66,142)( 67,144)( 68,143)( 69,137)( 70,138)( 71,140)( 72,139);;
s2 := (  1,  8)(  2,  6)(  3,  7)(  4,  5)(  9, 12)( 13, 32)( 14, 30)( 15, 31)
( 16, 29)( 17, 28)( 18, 26)( 19, 27)( 20, 25)( 21, 36)( 22, 34)( 23, 35)
( 24, 33)( 37, 44)( 38, 42)( 39, 43)( 40, 41)( 45, 48)( 49, 68)( 50, 66)
( 51, 67)( 52, 65)( 53, 64)( 54, 62)( 55, 63)( 56, 61)( 57, 72)( 58, 70)
( 59, 71)( 60, 69)( 73, 80)( 74, 78)( 75, 79)( 76, 77)( 81, 84)( 85,104)
( 86,102)( 87,103)( 88,101)( 89,100)( 90, 98)( 91, 99)( 92, 97)( 93,108)
( 94,106)( 95,107)( 96,105)(109,116)(110,114)(111,115)(112,113)(117,120)
(121,140)(122,138)(123,139)(124,137)(125,136)(126,134)(127,135)(128,133)
(129,144)(130,142)(131,143)(132,141);;
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*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(144)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)
( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)
( 38, 39)( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)
( 54, 67)( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 73,109)
( 74,111)( 75,110)( 76,112)( 77,113)( 78,115)( 79,114)( 80,116)( 81,117)
( 82,119)( 83,118)( 84,120)( 85,133)( 86,135)( 87,134)( 88,136)( 89,137)
( 90,139)( 91,138)( 92,140)( 93,141)( 94,143)( 95,142)( 96,144)( 97,121)
( 98,123)( 99,122)(100,124)(101,125)(102,127)(103,126)(104,128)(105,129)
(106,131)(107,130)(108,132);
s1 := Sym(144)!(  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 93)(  6, 94)(  7, 96)
(  8, 95)(  9, 89)( 10, 90)( 11, 92)( 12, 91)( 13, 73)( 14, 74)( 15, 76)
( 16, 75)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)
( 24, 79)( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,105)( 30,106)( 31,108)
( 32,107)( 33,101)( 34,102)( 35,104)( 36,103)( 37,121)( 38,122)( 39,124)
( 40,123)( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)
( 48,127)( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)
( 56,119)( 57,113)( 58,114)( 59,116)( 60,115)( 61,133)( 62,134)( 63,136)
( 64,135)( 65,141)( 66,142)( 67,144)( 68,143)( 69,137)( 70,138)( 71,140)
( 72,139);
s2 := Sym(144)!(  1,  8)(  2,  6)(  3,  7)(  4,  5)(  9, 12)( 13, 32)( 14, 30)
( 15, 31)( 16, 29)( 17, 28)( 18, 26)( 19, 27)( 20, 25)( 21, 36)( 22, 34)
( 23, 35)( 24, 33)( 37, 44)( 38, 42)( 39, 43)( 40, 41)( 45, 48)( 49, 68)
( 50, 66)( 51, 67)( 52, 65)( 53, 64)( 54, 62)( 55, 63)( 56, 61)( 57, 72)
( 58, 70)( 59, 71)( 60, 69)( 73, 80)( 74, 78)( 75, 79)( 76, 77)( 81, 84)
( 85,104)( 86,102)( 87,103)( 88,101)( 89,100)( 90, 98)( 91, 99)( 92, 97)
( 93,108)( 94,106)( 95,107)( 96,105)(109,116)(110,114)(111,115)(112,113)
(117,120)(121,140)(122,138)(123,139)(124,137)(125,136)(126,134)(127,135)
(128,133)(129,144)(130,142)(131,143)(132,141);
poly := sub<Sym(144)|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*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope