Polytope of Type {12,28}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope