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# Polytope of Type {2,6,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,12,4}*1152c
if this polytope has a name.
Group : SmallGroup(1152,134267)
Rank : 5
Schlafli Type : {2,6,12,4}
Number of vertices, edges, etc : 2, 6, 36, 24, 4
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {2,6,6,4}*576c, {2,6,12,2}*576c
3-fold quotients : {2,6,4,4}*384
4-fold quotients : {2,3,6,4}*288, {2,6,6,2}*288c
6-fold quotients : {2,6,2,4}*192, {2,6,4,2}*192a
8-fold quotients : {2,3,6,2}*144
9-fold quotients : {2,2,4,4}*128
12-fold quotients : {2,3,2,4}*96, {2,6,2,2}*96
18-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64
24-fold quotients : {2,3,2,2}*48
36-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)
(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)(125,127)
(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)(143,145);;
s2 := (  3, 79)(  4, 78)(  5, 80)(  6, 76)(  7, 75)(  8, 77)(  9, 82)( 10, 81)
( 11, 83)( 12, 88)( 13, 87)( 14, 89)( 15, 85)( 16, 84)( 17, 86)( 18, 91)
( 19, 90)( 20, 92)( 21, 97)( 22, 96)( 23, 98)( 24, 94)( 25, 93)( 26, 95)
( 27,100)( 28, 99)( 29,101)( 30,106)( 31,105)( 32,107)( 33,103)( 34,102)
( 35,104)( 36,109)( 37,108)( 38,110)( 39,115)( 40,114)( 41,116)( 42,112)
( 43,111)( 44,113)( 45,118)( 46,117)( 47,119)( 48,124)( 49,123)( 50,125)
( 51,121)( 52,120)( 53,122)( 54,127)( 55,126)( 56,128)( 57,133)( 58,132)
( 59,134)( 60,130)( 61,129)( 62,131)( 63,136)( 64,135)( 65,137)( 66,142)
( 67,141)( 68,143)( 69,139)( 70,138)( 71,140)( 72,145)( 73,144)( 74,146);;
s3 := (  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)( 25, 26)
( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 39, 57)( 40, 59)( 41, 58)( 42, 60)
( 43, 62)( 44, 61)( 45, 63)( 46, 65)( 47, 64)( 48, 66)( 49, 68)( 50, 67)
( 51, 69)( 52, 71)( 53, 70)( 54, 72)( 55, 74)( 56, 73)( 75, 84)( 76, 86)
( 77, 85)( 78, 87)( 79, 89)( 80, 88)( 81, 90)( 82, 92)( 83, 91)( 93,102)
( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)(101,109)
(111,138)(112,140)(113,139)(114,141)(115,143)(116,142)(117,144)(118,146)
(119,145)(120,129)(121,131)(122,130)(123,132)(124,134)(125,133)(126,135)
(127,137)(128,136);;
s4 := (  3, 39)(  4, 40)(  5, 41)(  6, 42)(  7, 43)(  8, 44)(  9, 45)( 10, 46)
( 11, 47)( 12, 48)( 13, 49)( 14, 50)( 15, 51)( 16, 52)( 17, 53)( 18, 54)
( 19, 55)( 20, 56)( 21, 57)( 22, 58)( 23, 59)( 24, 60)( 25, 61)( 26, 62)
( 27, 63)( 28, 64)( 29, 65)( 30, 66)( 31, 67)( 32, 68)( 33, 69)( 34, 70)
( 35, 71)( 36, 72)( 37, 73)( 38, 74)( 75,111)( 76,112)( 77,113)( 78,114)
( 79,115)( 80,116)( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)
( 87,123)( 88,124)( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)
( 95,131)( 96,132)( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)
(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(146)!(1,2);
s1 := Sym(146)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109)(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)
(125,127)(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)
(143,145);
s2 := Sym(146)!(  3, 79)(  4, 78)(  5, 80)(  6, 76)(  7, 75)(  8, 77)(  9, 82)
( 10, 81)( 11, 83)( 12, 88)( 13, 87)( 14, 89)( 15, 85)( 16, 84)( 17, 86)
( 18, 91)( 19, 90)( 20, 92)( 21, 97)( 22, 96)( 23, 98)( 24, 94)( 25, 93)
( 26, 95)( 27,100)( 28, 99)( 29,101)( 30,106)( 31,105)( 32,107)( 33,103)
( 34,102)( 35,104)( 36,109)( 37,108)( 38,110)( 39,115)( 40,114)( 41,116)
( 42,112)( 43,111)( 44,113)( 45,118)( 46,117)( 47,119)( 48,124)( 49,123)
( 50,125)( 51,121)( 52,120)( 53,122)( 54,127)( 55,126)( 56,128)( 57,133)
( 58,132)( 59,134)( 60,130)( 61,129)( 62,131)( 63,136)( 64,135)( 65,137)
( 66,142)( 67,141)( 68,143)( 69,139)( 70,138)( 71,140)( 72,145)( 73,144)
( 74,146);
s3 := Sym(146)!(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 19, 20)( 22, 23)
( 25, 26)( 28, 29)( 31, 32)( 34, 35)( 37, 38)( 39, 57)( 40, 59)( 41, 58)
( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 65)( 47, 64)( 48, 66)( 49, 68)
( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)( 55, 74)( 56, 73)( 75, 84)
( 76, 86)( 77, 85)( 78, 87)( 79, 89)( 80, 88)( 81, 90)( 82, 92)( 83, 91)
( 93,102)( 94,104)( 95,103)( 96,105)( 97,107)( 98,106)( 99,108)(100,110)
(101,109)(111,138)(112,140)(113,139)(114,141)(115,143)(116,142)(117,144)
(118,146)(119,145)(120,129)(121,131)(122,130)(123,132)(124,134)(125,133)
(126,135)(127,137)(128,136);
s4 := Sym(146)!(  3, 39)(  4, 40)(  5, 41)(  6, 42)(  7, 43)(  8, 44)(  9, 45)
( 10, 46)( 11, 47)( 12, 48)( 13, 49)( 14, 50)( 15, 51)( 16, 52)( 17, 53)
( 18, 54)( 19, 55)( 20, 56)( 21, 57)( 22, 58)( 23, 59)( 24, 60)( 25, 61)
( 26, 62)( 27, 63)( 28, 64)( 29, 65)( 30, 66)( 31, 67)( 32, 68)( 33, 69)
( 34, 70)( 35, 71)( 36, 72)( 37, 73)( 38, 74)( 75,111)( 76,112)( 77,113)
( 78,114)( 79,115)( 80,116)( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)
( 86,122)( 87,123)( 88,124)( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)
( 94,130)( 95,131)( 96,132)( 97,133)( 98,134)( 99,135)(100,136)(101,137)
(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(109,145)
(110,146);
poly := sub<Sym(146)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope