Polytope of Type {2,3,6,14}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,14}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 5
Schlafli Type : {2,3,6,14}
Number of vertices, edges, etc : 2, 4, 12, 56, 14
Order of s0s1s2s3s4 : 28
Order of s0s1s2s3s4s3s2s1 : 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) :
7-fold quotients : {2,3,6,2}*192
14-fold quotients : {2,3,3,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)( 31, 59)
( 32, 61)( 33, 60)( 34, 62)( 35, 63)( 36, 65)( 37, 64)( 38, 66)( 39, 67)
( 40, 69)( 41, 68)( 42, 70)( 43, 71)( 44, 73)( 45, 72)( 46, 74)( 47, 75)
( 48, 77)( 49, 76)( 50, 78)( 51, 79)( 52, 81)( 53, 80)( 54, 82)( 55, 83)
( 56, 85)( 57, 84)( 58, 86)( 88, 89)( 92, 93)( 96, 97)(100,101)(104,105)
(108,109)(112,113)(115,143)(116,145)(117,144)(118,146)(119,147)(120,149)
(121,148)(122,150)(123,151)(124,153)(125,152)(126,154)(127,155)(128,157)
(129,156)(130,158)(131,159)(132,161)(133,160)(134,162)(135,163)(136,165)
(137,164)(138,166)(139,167)(140,169)(141,168)(142,170);;
s2 := ( 3, 31)( 4, 32)( 5, 34)( 6, 33)( 7, 35)( 8, 36)( 9, 38)( 10, 37)
( 11, 39)( 12, 40)( 13, 42)( 14, 41)( 15, 43)( 16, 44)( 17, 46)( 18, 45)
( 19, 47)( 20, 48)( 21, 50)( 22, 49)( 23, 51)( 24, 52)( 25, 54)( 26, 53)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 61, 62)( 65, 66)( 69, 70)( 73, 74)
( 77, 78)( 81, 82)( 85, 86)( 87,115)( 88,116)( 89,118)( 90,117)( 91,119)
( 92,120)( 93,122)( 94,121)( 95,123)( 96,124)( 97,126)( 98,125)( 99,127)
(100,128)(101,130)(102,129)(103,131)(104,132)(105,134)(106,133)(107,135)
(108,136)(109,138)(110,137)(111,139)(112,140)(113,142)(114,141)(145,146)
(149,150)(153,154)(157,158)(161,162)(165,166)(169,170);;
s3 := ( 3, 6)( 7, 30)( 8, 28)( 9, 29)( 10, 27)( 11, 26)( 12, 24)( 13, 25)
( 14, 23)( 15, 22)( 16, 20)( 17, 21)( 18, 19)( 31, 62)( 32, 60)( 33, 61)
( 34, 59)( 35, 86)( 36, 84)( 37, 85)( 38, 83)( 39, 82)( 40, 80)( 41, 81)
( 42, 79)( 43, 78)( 44, 76)( 45, 77)( 46, 75)( 47, 74)( 48, 72)( 49, 73)
( 50, 71)( 51, 70)( 52, 68)( 53, 69)( 54, 67)( 55, 66)( 56, 64)( 57, 65)
( 58, 63)( 87, 90)( 91,114)( 92,112)( 93,113)( 94,111)( 95,110)( 96,108)
( 97,109)( 98,107)( 99,106)(100,104)(101,105)(102,103)(115,146)(116,144)
(117,145)(118,143)(119,170)(120,168)(121,169)(122,167)(123,166)(124,164)
(125,165)(126,163)(127,162)(128,160)(129,161)(130,159)(131,158)(132,156)
(133,157)(134,155)(135,154)(136,152)(137,153)(138,151)(139,150)(140,148)
(141,149)(142,147);;
s4 := ( 3, 91)( 4, 92)( 5, 93)( 6, 94)( 7, 87)( 8, 88)( 9, 89)( 10, 90)
( 11,111)( 12,112)( 13,113)( 14,114)( 15,107)( 16,108)( 17,109)( 18,110)
( 19,103)( 20,104)( 21,105)( 22,106)( 23, 99)( 24,100)( 25,101)( 26,102)
( 27, 95)( 28, 96)( 29, 97)( 30, 98)( 31,119)( 32,120)( 33,121)( 34,122)
( 35,115)( 36,116)( 37,117)( 38,118)( 39,139)( 40,140)( 41,141)( 42,142)
( 43,135)( 44,136)( 45,137)( 46,138)( 47,131)( 48,132)( 49,133)( 50,134)
( 51,127)( 52,128)( 53,129)( 54,130)( 55,123)( 56,124)( 57,125)( 58,126)
( 59,147)( 60,148)( 61,149)( 62,150)( 63,143)( 64,144)( 65,145)( 66,146)
( 67,167)( 68,168)( 69,169)( 70,170)( 71,163)( 72,164)( 73,165)( 74,166)
( 75,159)( 76,160)( 77,161)( 78,162)( 79,155)( 80,156)( 81,157)( 82,158)
( 83,151)( 84,152)( 85,153)( 86,154);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(170)!(1,2);
s1 := Sym(170)!( 4, 5)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)
( 31, 59)( 32, 61)( 33, 60)( 34, 62)( 35, 63)( 36, 65)( 37, 64)( 38, 66)
( 39, 67)( 40, 69)( 41, 68)( 42, 70)( 43, 71)( 44, 73)( 45, 72)( 46, 74)
( 47, 75)( 48, 77)( 49, 76)( 50, 78)( 51, 79)( 52, 81)( 53, 80)( 54, 82)
( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 88, 89)( 92, 93)( 96, 97)(100,101)
(104,105)(108,109)(112,113)(115,143)(116,145)(117,144)(118,146)(119,147)
(120,149)(121,148)(122,150)(123,151)(124,153)(125,152)(126,154)(127,155)
(128,157)(129,156)(130,158)(131,159)(132,161)(133,160)(134,162)(135,163)
(136,165)(137,164)(138,166)(139,167)(140,169)(141,168)(142,170);
s2 := Sym(170)!( 3, 31)( 4, 32)( 5, 34)( 6, 33)( 7, 35)( 8, 36)( 9, 38)
( 10, 37)( 11, 39)( 12, 40)( 13, 42)( 14, 41)( 15, 43)( 16, 44)( 17, 46)
( 18, 45)( 19, 47)( 20, 48)( 21, 50)( 22, 49)( 23, 51)( 24, 52)( 25, 54)
( 26, 53)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 61, 62)( 65, 66)( 69, 70)
( 73, 74)( 77, 78)( 81, 82)( 85, 86)( 87,115)( 88,116)( 89,118)( 90,117)
( 91,119)( 92,120)( 93,122)( 94,121)( 95,123)( 96,124)( 97,126)( 98,125)
( 99,127)(100,128)(101,130)(102,129)(103,131)(104,132)(105,134)(106,133)
(107,135)(108,136)(109,138)(110,137)(111,139)(112,140)(113,142)(114,141)
(145,146)(149,150)(153,154)(157,158)(161,162)(165,166)(169,170);
s3 := Sym(170)!( 3, 6)( 7, 30)( 8, 28)( 9, 29)( 10, 27)( 11, 26)( 12, 24)
( 13, 25)( 14, 23)( 15, 22)( 16, 20)( 17, 21)( 18, 19)( 31, 62)( 32, 60)
( 33, 61)( 34, 59)( 35, 86)( 36, 84)( 37, 85)( 38, 83)( 39, 82)( 40, 80)
( 41, 81)( 42, 79)( 43, 78)( 44, 76)( 45, 77)( 46, 75)( 47, 74)( 48, 72)
( 49, 73)( 50, 71)( 51, 70)( 52, 68)( 53, 69)( 54, 67)( 55, 66)( 56, 64)
( 57, 65)( 58, 63)( 87, 90)( 91,114)( 92,112)( 93,113)( 94,111)( 95,110)
( 96,108)( 97,109)( 98,107)( 99,106)(100,104)(101,105)(102,103)(115,146)
(116,144)(117,145)(118,143)(119,170)(120,168)(121,169)(122,167)(123,166)
(124,164)(125,165)(126,163)(127,162)(128,160)(129,161)(130,159)(131,158)
(132,156)(133,157)(134,155)(135,154)(136,152)(137,153)(138,151)(139,150)
(140,148)(141,149)(142,147);
s4 := Sym(170)!( 3, 91)( 4, 92)( 5, 93)( 6, 94)( 7, 87)( 8, 88)( 9, 89)
( 10, 90)( 11,111)( 12,112)( 13,113)( 14,114)( 15,107)( 16,108)( 17,109)
( 18,110)( 19,103)( 20,104)( 21,105)( 22,106)( 23, 99)( 24,100)( 25,101)
( 26,102)( 27, 95)( 28, 96)( 29, 97)( 30, 98)( 31,119)( 32,120)( 33,121)
( 34,122)( 35,115)( 36,116)( 37,117)( 38,118)( 39,139)( 40,140)( 41,141)
( 42,142)( 43,135)( 44,136)( 45,137)( 46,138)( 47,131)( 48,132)( 49,133)
( 50,134)( 51,127)( 52,128)( 53,129)( 54,130)( 55,123)( 56,124)( 57,125)
( 58,126)( 59,147)( 60,148)( 61,149)( 62,150)( 63,143)( 64,144)( 65,145)
( 66,146)( 67,167)( 68,168)( 69,169)( 70,170)( 71,163)( 72,164)( 73,165)
( 74,166)( 75,159)( 76,160)( 77,161)( 78,162)( 79,155)( 80,156)( 81,157)
( 82,158)( 83,151)( 84,152)( 85,153)( 86,154);
poly := sub<Sym(170)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope