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# Polytope of Type {6,12,4}

Atlas Canonical Name : {6,12,4}*576e
if this polytope has a name.
Group : SmallGroup(576,8312)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 6, 36, 24, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,12,4,2} of size 1152
Vertex Figure Of :
{2,6,12,4} of size 1152
{3,6,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,4}*288d
3-fold quotients : {2,12,4}*192c
6-fold quotients : {2,6,4}*96c
12-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12,4}*1152e, {6,12,4}*1152e
3-fold covers : {18,12,4}*1728d, {6,36,4}*1728d, {6,12,4}*1728g, {6,12,4}*1728m
Permutation Representation (GAP) :
```s0 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 29, 33)( 30, 34)( 31, 35)( 32, 36)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 69)( 66, 70)( 67, 71)( 68, 72)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
(101,105)(102,106)(103,107)(104,108)(113,117)(114,118)(115,119)(116,120)
(125,129)(126,130)(127,131)(128,132)(137,141)(138,142)(139,143)(140,144);;
s1 := (  1,  5)(  2,  7)(  3,  6)(  4,  8)( 10, 11)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 46, 47)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 73,113)( 74,115)( 75,114)( 76,116)( 77,109)( 78,111)
( 79,110)( 80,112)( 81,117)( 82,119)( 83,118)( 84,120)( 85,137)( 86,139)
( 87,138)( 88,140)( 89,133)( 90,135)( 91,134)( 92,136)( 93,141)( 94,143)
( 95,142)( 96,144)( 97,125)( 98,127)( 99,126)(100,128)(101,121)(102,123)
(103,122)(104,124)(105,129)(106,131)(107,130)(108,132);;
s2 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 89)(  6, 90)(  7, 92)(  8, 91)
(  9, 93)( 10, 94)( 11, 96)( 12, 95)( 13, 73)( 14, 74)( 15, 76)( 16, 75)
( 17, 77)( 18, 78)( 19, 80)( 20, 79)( 21, 81)( 22, 82)( 23, 84)( 24, 83)
( 25, 97)( 26, 98)( 27,100)( 28, 99)( 29,101)( 30,102)( 31,104)( 32,103)
( 33,105)( 34,106)( 35,108)( 36,107)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,125)( 42,126)( 43,128)( 44,127)( 45,129)( 46,130)( 47,132)( 48,131)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,113)( 54,114)( 55,116)( 56,115)
( 57,117)( 58,118)( 59,120)( 60,119)( 61,133)( 62,134)( 63,136)( 64,135)
( 65,137)( 66,138)( 67,140)( 68,139)( 69,141)( 70,142)( 71,144)( 72,143);;
s3 := (  1, 40)(  2, 39)(  3, 38)(  4, 37)(  5, 44)(  6, 43)(  7, 42)(  8, 41)
(  9, 48)( 10, 47)( 11, 46)( 12, 45)( 13, 52)( 14, 51)( 15, 50)( 16, 49)
( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 60)( 22, 59)( 23, 58)( 24, 57)
( 25, 64)( 26, 63)( 27, 62)( 28, 61)( 29, 68)( 30, 67)( 31, 66)( 32, 65)
( 33, 72)( 34, 71)( 35, 70)( 36, 69)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,116)( 78,115)( 79,114)( 80,113)( 81,120)( 82,119)( 83,118)( 84,117)
( 85,124)( 86,123)( 87,122)( 88,121)( 89,128)( 90,127)( 91,126)( 92,125)
( 93,132)( 94,131)( 95,130)( 96,129)( 97,136)( 98,135)( 99,134)(100,133)
(101,140)(102,139)(103,138)(104,137)(105,144)(106,143)(107,142)(108,141);;
poly := Group([s0,s1,s2,s3]);;

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

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

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

```
References : None.
to this polytope