Polytope of Type {2,28,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,28,12}*1344
if this polytope has a name.
Group : SmallGroup(1344,9160)
Rank : 4
Schlafli Type : {2,28,12}
Number of vertices, edges, etc : 2, 28, 168, 12
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,14,12}*672, {2,28,6}*672a
   3-fold quotients : {2,28,4}*448
   4-fold quotients : {2,14,6}*336
   6-fold quotients : {2,28,2}*224, {2,14,4}*224
   7-fold quotients : {2,4,12}*192a
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {2,2,12}*96, {2,4,6}*96a
   21-fold quotients : {2,4,4}*64
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {2,2,6}*48
   42-fold quotients : {2,2,4}*32, {2,4,2}*32
   56-fold quotients : {2,2,3}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  9)(  5,  8)(  6,  7)( 11, 16)( 12, 15)( 13, 14)( 18, 23)( 19, 22)
( 20, 21)( 25, 30)( 26, 29)( 27, 28)( 32, 37)( 33, 36)( 34, 35)( 39, 44)
( 40, 43)( 41, 42)( 46, 51)( 47, 50)( 48, 49)( 53, 58)( 54, 57)( 55, 56)
( 60, 65)( 61, 64)( 62, 63)( 67, 72)( 68, 71)( 69, 70)( 74, 79)( 75, 78)
( 76, 77)( 81, 86)( 82, 85)( 83, 84)( 87,129)( 88,135)( 89,134)( 90,133)
( 91,132)( 92,131)( 93,130)( 94,136)( 95,142)( 96,141)( 97,140)( 98,139)
( 99,138)(100,137)(101,143)(102,149)(103,148)(104,147)(105,146)(106,145)
(107,144)(108,150)(109,156)(110,155)(111,154)(112,153)(113,152)(114,151)
(115,157)(116,163)(117,162)(118,161)(119,160)(120,159)(121,158)(122,164)
(123,170)(124,169)(125,168)(126,167)(127,166)(128,165);;
s2 := (  3, 88)(  4, 87)(  5, 93)(  6, 92)(  7, 91)(  8, 90)(  9, 89)( 10,102)
( 11,101)( 12,107)( 13,106)( 14,105)( 15,104)( 16,103)( 17, 95)( 18, 94)
( 19,100)( 20, 99)( 21, 98)( 22, 97)( 23, 96)( 24,109)( 25,108)( 26,114)
( 27,113)( 28,112)( 29,111)( 30,110)( 31,123)( 32,122)( 33,128)( 34,127)
( 35,126)( 36,125)( 37,124)( 38,116)( 39,115)( 40,121)( 41,120)( 42,119)
( 43,118)( 44,117)( 45,130)( 46,129)( 47,135)( 48,134)( 49,133)( 50,132)
( 51,131)( 52,144)( 53,143)( 54,149)( 55,148)( 56,147)( 57,146)( 58,145)
( 59,137)( 60,136)( 61,142)( 62,141)( 63,140)( 64,139)( 65,138)( 66,151)
( 67,150)( 68,156)( 69,155)( 70,154)( 71,153)( 72,152)( 73,165)( 74,164)
( 75,170)( 76,169)( 77,168)( 78,167)( 79,166)( 80,158)( 81,157)( 82,163)
( 83,162)( 84,161)( 85,160)( 86,159);;
s3 := (  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)(  8, 15)(  9, 16)( 24, 31)
( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 29, 36)( 30, 37)( 45, 52)( 46, 53)
( 47, 54)( 48, 55)( 49, 56)( 50, 57)( 51, 58)( 66, 73)( 67, 74)( 68, 75)
( 69, 76)( 70, 77)( 71, 78)( 72, 79)( 87,115)( 88,116)( 89,117)( 90,118)
( 91,119)( 92,120)( 93,121)( 94,108)( 95,109)( 96,110)( 97,111)( 98,112)
( 99,113)(100,114)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)
(107,128)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)
(136,150)(137,151)(138,152)(139,153)(140,154)(141,155)(142,156)(143,164)
(144,165)(145,166)(146,167)(147,168)(148,169)(149,170);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(170)!(1,2);
s1 := Sym(170)!(  4,  9)(  5,  8)(  6,  7)( 11, 16)( 12, 15)( 13, 14)( 18, 23)
( 19, 22)( 20, 21)( 25, 30)( 26, 29)( 27, 28)( 32, 37)( 33, 36)( 34, 35)
( 39, 44)( 40, 43)( 41, 42)( 46, 51)( 47, 50)( 48, 49)( 53, 58)( 54, 57)
( 55, 56)( 60, 65)( 61, 64)( 62, 63)( 67, 72)( 68, 71)( 69, 70)( 74, 79)
( 75, 78)( 76, 77)( 81, 86)( 82, 85)( 83, 84)( 87,129)( 88,135)( 89,134)
( 90,133)( 91,132)( 92,131)( 93,130)( 94,136)( 95,142)( 96,141)( 97,140)
( 98,139)( 99,138)(100,137)(101,143)(102,149)(103,148)(104,147)(105,146)
(106,145)(107,144)(108,150)(109,156)(110,155)(111,154)(112,153)(113,152)
(114,151)(115,157)(116,163)(117,162)(118,161)(119,160)(120,159)(121,158)
(122,164)(123,170)(124,169)(125,168)(126,167)(127,166)(128,165);
s2 := Sym(170)!(  3, 88)(  4, 87)(  5, 93)(  6, 92)(  7, 91)(  8, 90)(  9, 89)
( 10,102)( 11,101)( 12,107)( 13,106)( 14,105)( 15,104)( 16,103)( 17, 95)
( 18, 94)( 19,100)( 20, 99)( 21, 98)( 22, 97)( 23, 96)( 24,109)( 25,108)
( 26,114)( 27,113)( 28,112)( 29,111)( 30,110)( 31,123)( 32,122)( 33,128)
( 34,127)( 35,126)( 36,125)( 37,124)( 38,116)( 39,115)( 40,121)( 41,120)
( 42,119)( 43,118)( 44,117)( 45,130)( 46,129)( 47,135)( 48,134)( 49,133)
( 50,132)( 51,131)( 52,144)( 53,143)( 54,149)( 55,148)( 56,147)( 57,146)
( 58,145)( 59,137)( 60,136)( 61,142)( 62,141)( 63,140)( 64,139)( 65,138)
( 66,151)( 67,150)( 68,156)( 69,155)( 70,154)( 71,153)( 72,152)( 73,165)
( 74,164)( 75,170)( 76,169)( 77,168)( 78,167)( 79,166)( 80,158)( 81,157)
( 82,163)( 83,162)( 84,161)( 85,160)( 86,159);
s3 := Sym(170)!(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)(  8, 15)(  9, 16)
( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 29, 36)( 30, 37)( 45, 52)
( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 50, 57)( 51, 58)( 66, 73)( 67, 74)
( 68, 75)( 69, 76)( 70, 77)( 71, 78)( 72, 79)( 87,115)( 88,116)( 89,117)
( 90,118)( 91,119)( 92,120)( 93,121)( 94,108)( 95,109)( 96,110)( 97,111)
( 98,112)( 99,113)(100,114)(101,122)(102,123)(103,124)(104,125)(105,126)
(106,127)(107,128)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)
(135,163)(136,150)(137,151)(138,152)(139,153)(140,154)(141,155)(142,156)
(143,164)(144,165)(145,166)(146,167)(147,168)(148,169)(149,170);
poly := sub<Sym(170)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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 >;

to this polytope