Polytope of Type {28,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,30}*1680b
if this polytope has a name.
Group : SmallGroup(1680,952)
Rank : 3
Schlafli Type : {28,30}
Number of vertices, edges, etc : 28, 420, 30
Order of s₀s₁s₂ : 105
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   5-fold quotients : {28,6}*336b
   7-fold quotients : {4,30}*240c
   14-fold quotients : {4,15}*120
   35-fold quotients : {4,6}*48b
   70-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope