Polytope of Type {30,28}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope