Polytope of Type {2,10,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,28}*1120
if this polytope has a name.
Group : SmallGroup(1120,989)
Rank : 4
Schlafli Type : {2,10,28}
Number of vertices, edges, etc : 2, 10, 140, 28
Order of s₀s₁s₂s₃ : 140
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 : {2,10,14}*560
   5-fold quotients : {2,2,28}*224
   7-fold quotients : {2,10,4}*160
   10-fold quotients : {2,2,14}*112
   14-fold quotients : {2,10,2}*80
   20-fold quotients : {2,2,7}*56
   28-fold quotients : {2,5,2}*40
   35-fold quotients : {2,2,4}*32
   70-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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