Polytope of Type {28,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,12}*672
Also Known As : {28,12|2}. if this polytope has another name.
Group : SmallGroup(672,620)
Rank : 3
Schlafli Type : {28,12}
Number of vertices, edges, etc : 28, 168, 12
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {28,12,2} of size 1344
Vertex Figure Of :
   {2,28,12} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {14,12}*336, {28,6}*336a
   3-fold quotients : {28,4}*224
   4-fold quotients : {14,6}*168
   6-fold quotients : {28,2}*112, {14,4}*112
   7-fold quotients : {4,12}*96a
   12-fold quotients : {14,2}*56
   14-fold quotients : {2,12}*48, {4,6}*48a
   21-fold quotients : {4,4}*32
   24-fold quotients : {7,2}*28
   28-fold quotients : {2,6}*24
   42-fold quotients : {2,4}*16, {4,2}*16
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {28,12}*1344a, {28,24}*1344a, {56,12}*1344a, {28,24}*1344b, {56,12}*1344b
Permutation Representation (GAP) :

s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 85,127)( 86,133)( 87,132)( 88,131)
( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)( 96,137)
( 97,136)( 98,135)( 99,141)(100,147)(101,146)(102,145)(103,144)(104,143)
(105,142)(106,148)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)
(113,155)(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)(120,162)
(121,168)(122,167)(123,166)(124,165)(125,164)(126,163);;
s1 := (  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)(  8,100)
(  9, 99)( 10,105)( 11,104)( 12,103)( 13,102)( 14,101)( 15, 93)( 16, 92)
( 17, 98)( 18, 97)( 19, 96)( 20, 95)( 21, 94)( 22,107)( 23,106)( 24,112)
( 25,111)( 26,110)( 27,109)( 28,108)( 29,121)( 30,120)( 31,126)( 32,125)
( 33,124)( 34,123)( 35,122)( 36,114)( 37,113)( 38,119)( 39,118)( 40,117)
( 41,116)( 42,115)( 43,128)( 44,127)( 45,133)( 46,132)( 47,131)( 48,130)
( 49,129)( 50,142)( 51,141)( 52,147)( 53,146)( 54,145)( 55,144)( 56,143)
( 57,135)( 58,134)( 59,140)( 60,139)( 61,138)( 62,137)( 63,136)( 64,149)
( 65,148)( 66,154)( 67,153)( 68,152)( 69,151)( 70,150)( 71,163)( 72,162)
( 73,168)( 74,167)( 75,166)( 76,165)( 77,164)( 78,156)( 79,155)( 80,161)
( 81,160)( 82,159)( 83,158)( 84,157);;
s2 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 22, 29)
( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85,113)( 86,114)( 87,115)( 88,116)
( 89,117)( 90,118)( 91,119)( 92,106)( 93,107)( 94,108)( 95,109)( 96,110)
( 97,111)( 98,112)( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)
(105,126)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)
(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)(141,162)
(142,163)(143,164)(144,165)(145,166)(146,167)(147,168);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(168)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 85,127)( 86,133)( 87,132)
( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)
( 96,137)( 97,136)( 98,135)( 99,141)(100,147)(101,146)(102,145)(103,144)
(104,143)(105,142)(106,148)(107,154)(108,153)(109,152)(110,151)(111,150)
(112,149)(113,155)(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)
(120,162)(121,168)(122,167)(123,166)(124,165)(125,164)(126,163);
s1 := Sym(168)!(  1, 86)(  2, 85)(  3, 91)(  4, 90)(  5, 89)(  6, 88)(  7, 87)
(  8,100)(  9, 99)( 10,105)( 11,104)( 12,103)( 13,102)( 14,101)( 15, 93)
( 16, 92)( 17, 98)( 18, 97)( 19, 96)( 20, 95)( 21, 94)( 22,107)( 23,106)
( 24,112)( 25,111)( 26,110)( 27,109)( 28,108)( 29,121)( 30,120)( 31,126)
( 32,125)( 33,124)( 34,123)( 35,122)( 36,114)( 37,113)( 38,119)( 39,118)
( 40,117)( 41,116)( 42,115)( 43,128)( 44,127)( 45,133)( 46,132)( 47,131)
( 48,130)( 49,129)( 50,142)( 51,141)( 52,147)( 53,146)( 54,145)( 55,144)
( 56,143)( 57,135)( 58,134)( 59,140)( 60,139)( 61,138)( 62,137)( 63,136)
( 64,149)( 65,148)( 66,154)( 67,153)( 68,152)( 69,151)( 70,150)( 71,163)
( 72,162)( 73,168)( 74,167)( 75,166)( 76,165)( 77,164)( 78,156)( 79,155)
( 80,161)( 81,160)( 82,159)( 83,158)( 84,157);
s2 := Sym(168)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 22, 29)( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85,113)( 86,114)( 87,115)
( 88,116)( 89,117)( 90,118)( 91,119)( 92,106)( 93,107)( 94,108)( 95,109)
( 96,110)( 97,111)( 98,112)( 99,120)(100,121)(101,122)(102,123)(103,124)
(104,125)(105,126)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)
(133,161)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)
(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168);
poly := sub<Sym(168)|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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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 >;

References : None.
to this polytope