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# Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*576f
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 : 12
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 : {12,6}*288b
3-fold quotients : {4,6}*192b
4-fold quotients : {12,6}*144c, {12,3}*144
6-fold quotients : {4,6}*96
8-fold quotients : {6,6}*72b
12-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {6,3}*36
24-fold quotients : {4,3}*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}*1152o, {24,6}*1152k, {24,6}*1152l, {12,12}*1152r, {12,6}*1152f
3-fold covers : {12,18}*1728d, {12,6}*1728f, {12,6}*1728i
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 15)( 14, 16)
( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 27)( 26, 28)( 29, 35)( 30, 36)
( 31, 33)( 32, 34)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 51)( 50, 52)( 53, 59)( 54, 60)( 55, 57)( 56, 58)( 61, 63)( 62, 64)
( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73,111)( 74,112)( 75,109)( 76,110)
( 77,119)( 78,120)( 79,117)( 80,118)( 81,115)( 82,116)( 83,113)( 84,114)
( 85,123)( 86,124)( 87,121)( 88,122)( 89,131)( 90,132)( 91,129)( 92,130)
( 93,127)( 94,128)( 95,125)( 96,126)( 97,135)( 98,136)( 99,133)(100,134)
(101,143)(102,144)(103,141)(104,142)(105,139)(106,140)(107,137)(108,138);;
s1 := (  1, 77)(  2, 79)(  3, 78)(  4, 80)(  5, 73)(  6, 75)(  7, 74)(  8, 76)
(  9, 81)( 10, 83)( 11, 82)( 12, 84)( 13,101)( 14,103)( 15,102)( 16,104)
( 17, 97)( 18, 99)( 19, 98)( 20,100)( 21,105)( 22,107)( 23,106)( 24,108)
( 25, 89)( 26, 91)( 27, 90)( 28, 92)( 29, 85)( 30, 87)( 31, 86)( 32, 88)
( 33, 93)( 34, 95)( 35, 94)( 36, 96)( 37,113)( 38,115)( 39,114)( 40,116)
( 41,109)( 42,111)( 43,110)( 44,112)( 45,117)( 46,119)( 47,118)( 48,120)
( 49,137)( 50,139)( 51,138)( 52,140)( 53,133)( 54,135)( 55,134)( 56,136)
( 57,141)( 58,143)( 59,142)( 60,144)( 61,125)( 62,127)( 63,126)( 64,128)
( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132);;
s2 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 33)(  6, 36)(  7, 35)(  8, 34)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 14, 16)( 17, 21)( 18, 24)( 19, 23)
( 20, 22)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 69)( 42, 72)( 43, 71)
( 44, 70)( 45, 65)( 46, 68)( 47, 67)( 48, 66)( 50, 52)( 53, 57)( 54, 60)
( 55, 59)( 56, 58)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,105)( 78,108)
( 79,107)( 80,106)( 81,101)( 82,104)( 83,103)( 84,102)( 86, 88)( 89, 93)
( 90, 96)( 91, 95)( 92, 94)(109,133)(110,136)(111,135)(112,134)(113,141)
(114,144)(115,143)(116,142)(117,137)(118,140)(119,139)(120,138)(122,124)
(125,129)(126,132)(127,131)(128,130);;
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*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope