Polytope of Type {12,4} with group SmallGroup(768,1087743)

Overview

Group: SmallGroup(768,1087743)
Rank: 3
Schläfli Type: {12,4}
Vertices, edges, …: 96, 192, 32
Order of s₀s₁s₂: 12
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie

Quotients maximal quotients in bold

2-fold

{12,4}*384d

4-fold

8-fold

12-fold

{4,4}*64

16-fold

24-fold

{4,4}*32

32-fold

48-fold

64-fold

{3,2}*12

96-fold

{2,2}*8

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*s0*s1*s2*s1*s0*s2*s1> of order 2

16 facets

16 of {12}*24

56 vertex figures

40 of {4}*8
16 of {2}*4

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

16 facets

16 of {12}*24

48 vertex figures

48 of {4}*8

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

16 facets

16 of {12}*24

48 vertex figures

48 of {4}*8

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

16 facets

16 of {12}*24

48 vertex figures

48 of {4}*8

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

16 facets

16 of {12}*24

48 vertex figures

48 of {4}*8

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

16 facets

16 of {12}*24

48 vertex figures

48 of {4}*8

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

16 facets

16 of {12}*24

48 vertex figures

48 of {4}*8

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

24 facets

16 of {6}*12
8 of {12}*24

48 vertex figures

48 of {4}*8

P/N, where N=<(s0*s1)^4> of order 3

16 facets

8 of {4}*8
8 of {12}*24

32 vertex figures

32 of {4}*8

P/N, where N=<s0*s1*s2*s1*s0*s2, (s0*s1)^3*s2*s1*s0*s2*(s1*s0)^2> of order 4

8 facets

8 of {12}*24

32 vertex figures

16 of {4}*8
16 of {2}*4

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

8 facets

8 of {12}*24

24 vertex figures

24 of {4}*8

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

8 facets

8 of {12}*24

28 vertex figures

20 of {4}*8
8 of {2}*4

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

8 facets

8 of {12}*24

24 vertex figures

24 of {4}*8

P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^3*s1*s2, s0*s2*(s1*s0)^4*s2*s1*s0*s1> of order 4

8 facets

8 of {12}*24

24 vertex figures

24 of {4}*8

P/N, where N=<(s0*s1)^6, s0*s1*s0*s2*(s1*s0)^4*s2*s1> of order 4

12 facets

8 of {6}*12
4 of {12}*24

24 vertex figures

24 of {4}*8

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

8 facets

8 of {12}*24

24 vertex figures

24 of {4}*8

P/N, where N=<(s0*s1)^6, (s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1> of order 4

12 facets

8 of {6}*12
4 of {12}*24

24 vertex figures

24 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^3*s0*s2*(s1*s0)^2*s2*s1> of order 4

8 facets

8 of {12}*24

24 vertex figures

24 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^6> of order 4

12 facets

8 of {6}*12
4 of {12}*24

24 vertex figures

24 of {4}*8

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, (s0*s1)^3*s0*s2*(s1*s0)^2*s1*s2> of order 4

8 facets

8 of {12}*24

28 vertex figures

20 of {4}*8
8 of {2}*4

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

8 facets

8 of {12}*24

32 vertex figures

16 of {2}*4
16 of {4}*8

P/N, where N=<(s0*s1)^3> of order 4

16 facets

8 of {3}*6
4 of {12}*24
4 of {6}*12

24 vertex figures

24 of {4}*8

P/N, where N=<(s0*s2*s1)^3> of order 4

8 facets

8 of {12}*24

24 vertex figures

24 of {4}*8

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

8 facets

8 of {12}*24

28 vertex figures

20 of {4}*8
8 of {2}*4

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

12 facets

4 of {2}*4
2 of {4}*8
2 of {12}*24
4 of {6}*12

16 vertex figures

16 of {4}*8

P/N, where N=<(s0*s1)^6, (s1*s0)^2*s2*(s1*s0)^3*s1*s2> of order 8

6 facets

4 of {6}*12
2 of {12}*24

12 vertex figures

12 of {4}*8

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s2, (s0*s1)^6> of order 8

6 facets

4 of {6}*12
2 of {12}*24

12 vertex figures

12 of {4}*8

P/N, where N=<(s1*s2)^2, (s1*s0*s1*s2)^2, (s0*s1)^2*s2*(s1*s0)^2*s2> of order 8

4 facets

4 of {12}*24

18 vertex figures

12 of {2}*4
6 of {4}*8

Representations

Permutation Representation (GAP)

s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 73, 85)( 74, 86)( 75, 88)( 76, 87)( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 97,121)( 98,122)( 99,124)(100,123)(101,129)(102,130)(103,132)(104,131)(105,125)(106,126)(107,128)(108,127)(109,133)(110,134)(111,136)(112,135)(113,141)(114,142)(115,144)(116,143)(117,137)(118,138)(119,140)(120,139)(145,181)(146,182)(147,184)(148,183)(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)(163,180)(164,179)(165,173)(166,174)(167,176)(168,175);;
s1 := (  1,129)(  2,131)(  3,130)(  4,132)(  5,125)(  6,127)(  7,126)(  8,128)(  9,121)( 10,123)( 11,122)( 12,124)( 13,141)( 14,143)( 15,142)( 16,144)( 17,137)( 18,139)( 19,138)( 20,140)( 21,133)( 22,135)( 23,134)( 24,136)( 25,105)( 26,107)( 27,106)( 28,108)( 29,101)( 30,103)( 31,102)( 32,104)( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,117)( 38,119)( 39,118)( 40,120)( 41,113)( 42,115)( 43,114)( 44,116)( 45,109)( 46,111)( 47,110)( 48,112)( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)( 56,176)( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)( 64,192)( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)( 72,184)( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)( 80,152)( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)( 88,168)( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)( 96,160);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 38)( 26, 37)( 27, 40)( 28, 39)( 29, 42)( 30, 41)( 31, 44)( 32, 43)( 33, 46)( 34, 45)( 35, 48)( 36, 47)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 86)( 74, 85)( 75, 88)( 76, 87)( 77, 90)( 78, 89)( 79, 92)( 80, 91)( 81, 94)( 82, 93)( 83, 96)( 84, 95)( 97,146)( 98,145)( 99,148)(100,147)(101,150)(102,149)(103,152)(104,151)(105,154)(106,153)(107,156)(108,155)(109,158)(110,157)(111,160)(112,159)(113,162)(114,161)(115,164)(116,163)(117,166)(118,165)(119,168)(120,167)(121,182)(122,181)(123,184)(124,183)(125,186)(126,185)(127,188)(128,187)(129,190)(130,189)(131,192)(132,191)(133,170)(134,169)(135,172)(136,171)(137,174)(138,173)(139,176)(140,175)(141,178)(142,177)(143,180)(144,179);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(192)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 69)( 54, 70)( 55, 72)( 56, 71)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 73, 85)( 74, 86)( 75, 88)( 76, 87)( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)( 97,121)( 98,122)( 99,124)(100,123)(101,129)(102,130)(103,132)(104,131)(105,125)(106,126)(107,128)(108,127)(109,133)(110,134)(111,136)(112,135)(113,141)(114,142)(115,144)(116,143)(117,137)(118,138)(119,140)(120,139)(145,181)(146,182)(147,184)(148,183)(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,169)(158,170)(159,172)(160,171)(161,177)(162,178)(163,180)(164,179)(165,173)(166,174)(167,176)(168,175);
s1 := Sym(192)!(  1,129)(  2,131)(  3,130)(  4,132)(  5,125)(  6,127)(  7,126)(  8,128)(  9,121)( 10,123)( 11,122)( 12,124)( 13,141)( 14,143)( 15,142)( 16,144)( 17,137)( 18,139)( 19,138)( 20,140)( 21,133)( 22,135)( 23,134)( 24,136)( 25,105)( 26,107)( 27,106)( 28,108)( 29,101)( 30,103)( 31,102)( 32,104)( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,117)( 38,119)( 39,118)( 40,120)( 41,113)( 42,115)( 43,114)( 44,116)( 45,109)( 46,111)( 47,110)( 48,112)( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)( 56,176)( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)( 64,192)( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)( 72,184)( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)( 80,152)( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)( 88,168)( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)( 96,160);
s2 := Sym(192)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 38)( 26, 37)( 27, 40)( 28, 39)( 29, 42)( 30, 41)( 31, 44)( 32, 43)( 33, 46)( 34, 45)( 35, 48)( 36, 47)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 86)( 74, 85)( 75, 88)( 76, 87)( 77, 90)( 78, 89)( 79, 92)( 80, 91)( 81, 94)( 82, 93)( 83, 96)( 84, 95)( 97,146)( 98,145)( 99,148)(100,147)(101,150)(102,149)(103,152)(104,151)(105,154)(106,153)(107,156)(108,155)(109,158)(110,157)(111,160)(112,159)(113,162)(114,161)(115,164)(116,163)(117,166)(118,165)(119,168)(120,167)(121,182)(122,181)(123,184)(124,183)(125,186)(126,185)(127,188)(128,187)(129,190)(130,189)(131,192)(132,191)(133,170)(134,169)(135,172)(136,171)(137,174)(138,173)(139,176)(140,175)(141,178)(142,177)(143,180)(144,179);
poly := sub<Sym(192)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.

Overview

Special Properties

Related Polytopes

Quotients maximal quotients in bold

Covers minimal covers in bold

Irregular Quotients of which this is a minimal cover

Representations

References

Twisty Puzzle