Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*1296o

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Overview

Group
SmallGroup(1296,2977)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
54, 324, 108
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

108-fold

162-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s2*s1*(s0*(s2*s1)^2)^2*s2> of order 2

54 facets

30 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2> of order 3

36 facets

18 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1> of order 3

48 facets

18 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 3

36 facets

18 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^2*(s1*s2)^2> of order 4

27 facets

15 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*(s2*s1)^2*s0*(s2*s1)^3*s2> of order 6

24 facets

12 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s1*s2*s1*(s0*(s2*s1)^2)^2*s2> of order 6

18 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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