Polytope of Type {2,12,28}

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

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

Permutation Representation (Magma) :

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

to this polytope