Polytope of Type {12,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*576b
if this polytope has a name.
Group : SmallGroup(576,2480)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 12, 144, 24
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
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}*288c, {6,24}*288c
   3-fold quotients : {12,8}*192a
   4-fold quotients : {12,6}*144b, {6,12}*144c
   6-fold quotients : {12,4}*96a, {6,8}*96
   8-fold quotients : {6,6}*72c
   9-fold quotients : {4,8}*64a
   12-fold quotients : {12,2}*48, {6,4}*48a
   16-fold quotients : {3,6}*36
   18-fold quotients : {4,4}*32, {2,8}*32
   24-fold quotients : {6,2}*24
   36-fold quotients : {2,4}*16, {4,2}*16
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,24}*1152c, {24,24}*1152d, {24,24}*1152e, {12,48}*1152c, {12,48}*1152f
   3-fold covers : {36,24}*1728b, {12,24}*1728b, {12,24}*1728o
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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