Questions?
See the FAQ
or other info.

# Polytope of Type {12,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*576e
if this polytope has a name.
Group : SmallGroup(576,2897)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 12, 144, 24
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,24,2} of size 1152
Vertex Figure Of :
{2,12,24} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,12}*288a
3-fold quotients : {4,24}*192b, {12,8}*192b
4-fold quotients : {6,12}*144a, {12,6}*144a
6-fold quotients : {4,12}*96a, {12,4}*96a
8-fold quotients : {6,6}*72a
9-fold quotients : {4,8}*64b
12-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
18-fold quotients : {4,4}*32
24-fold quotients : {2,6}*24, {6,2}*24
36-fold quotients : {2,4}*16, {4,2}*16
48-fold quotients : {2,3}*12, {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,24}*1152b, {24,24}*1152g, {24,24}*1152k
3-fold covers : {12,72}*1728c, {36,24}*1728d, {12,24}*1728f, {12,24}*1728p
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)( 20, 29)
( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 40, 43)
( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 55, 64)( 56, 65)( 57, 66)
( 58, 70)( 59, 71)( 60, 72)( 61, 67)( 62, 68)( 63, 69)( 73,109)( 74,110)
( 75,111)( 76,115)( 77,116)( 78,117)( 79,112)( 80,113)( 81,114)( 82,118)
( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)( 88,121)( 89,122)( 90,123)
( 91,136)( 92,137)( 93,138)( 94,142)( 95,143)( 96,144)( 97,139)( 98,140)
( 99,141)(100,127)(101,128)(102,129)(103,133)(104,134)(105,135)(106,130)
(107,131)(108,132);;
s1 := (  1, 76)(  2, 78)(  3, 77)(  4, 73)(  5, 75)(  6, 74)(  7, 79)(  8, 81)
(  9, 80)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)( 16, 88)
( 17, 90)( 18, 89)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)( 24,101)
( 25,106)( 26,108)( 27,107)( 28, 94)( 29, 96)( 30, 95)( 31, 91)( 32, 93)
( 33, 92)( 34, 97)( 35, 99)( 36, 98)( 37,112)( 38,114)( 39,113)( 40,109)
( 41,111)( 42,110)( 43,115)( 44,117)( 45,116)( 46,121)( 47,123)( 48,122)
( 49,118)( 50,120)( 51,119)( 52,124)( 53,126)( 54,125)( 55,139)( 56,141)
( 57,140)( 58,136)( 59,138)( 60,137)( 61,142)( 62,144)( 63,143)( 64,130)
( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)( 72,134);;
s2 := (  1,  2)(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 19, 29)( 20, 28)
( 21, 30)( 22, 32)( 23, 31)( 24, 33)( 25, 35)( 26, 34)( 27, 36)( 37, 47)
( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)
( 55, 56)( 58, 59)( 61, 62)( 64, 65)( 67, 68)( 70, 71)( 73, 92)( 74, 91)
( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 98)( 80, 97)( 81, 99)( 82,101)
( 83,100)( 84,102)( 85,104)( 86,103)( 87,105)( 88,107)( 89,106)( 90,108)
(109,137)(110,136)(111,138)(112,140)(113,139)(114,141)(115,143)(116,142)
(117,144)(118,128)(119,127)(120,129)(121,131)(122,130)(123,132)(124,134)
(125,133)(126,135);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*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*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope