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# Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*768f
if this polytope has a name.
Group : SmallGroup(768,1087581)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 32, 192, 64
Order of s0s1s2 : 4
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*384c
4-fold quotients : {6,6}*192a, {6,12}*192a
8-fold quotients : {6,6}*96
16-fold quotients : {3,6}*48, {6,3}*48
32-fold quotients : {3,3}*24
48-fold quotients : {2,4}*16
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(103,104)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,133)(118,134)
(119,136)(120,135)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(147,148)(151,152)(153,157)(154,158)(155,160)(156,159)
(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)
(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s1 := (  1,129)(  2,131)(  3,130)(  4,132)(  5,139)(  6,137)(  7,140)(  8,138)
(  9,134)( 10,136)( 11,133)( 12,135)( 13,144)( 14,142)( 15,143)( 16,141)
( 17,113)( 18,115)( 19,114)( 20,116)( 21,123)( 22,121)( 23,124)( 24,122)
( 25,118)( 26,120)( 27,117)( 28,119)( 29,128)( 30,126)( 31,127)( 32,125)
( 33, 97)( 34, 99)( 35, 98)( 36,100)( 37,107)( 38,105)( 39,108)( 40,106)
( 41,102)( 42,104)( 43,101)( 44,103)( 45,112)( 46,110)( 47,111)( 48,109)
( 49,177)( 50,179)( 51,178)( 52,180)( 53,187)( 54,185)( 55,188)( 56,186)
( 57,182)( 58,184)( 59,181)( 60,183)( 61,192)( 62,190)( 63,191)( 64,189)
( 65,161)( 66,163)( 67,162)( 68,164)( 69,171)( 70,169)( 71,172)( 72,170)
( 73,166)( 74,168)( 75,165)( 76,167)( 77,176)( 78,174)( 79,175)( 80,173)
( 81,145)( 82,147)( 83,146)( 84,148)( 85,155)( 86,153)( 87,156)( 88,154)
( 89,150)( 90,152)( 91,149)( 92,151)( 93,160)( 94,158)( 95,159)( 96,157);;
s2 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)( 11, 12)( 15, 16)( 17, 37)( 18, 38)
( 19, 40)( 20, 39)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 25, 41)( 26, 42)
( 27, 44)( 28, 43)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49, 53)( 50, 54)
( 51, 56)( 52, 55)( 59, 60)( 63, 64)( 65, 85)( 66, 86)( 67, 88)( 68, 87)
( 69, 81)( 70, 82)( 71, 84)( 72, 83)( 73, 89)( 74, 90)( 75, 92)( 76, 91)
( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 97,149)( 98,150)( 99,152)(100,151)
(101,145)(102,146)(103,148)(104,147)(105,153)(106,154)(107,156)(108,155)
(109,157)(110,158)(111,160)(112,159)(113,181)(114,182)(115,184)(116,183)
(117,177)(118,178)(119,180)(120,179)(121,185)(122,186)(123,188)(124,187)
(125,189)(126,190)(127,192)(128,191)(129,165)(130,166)(131,168)(132,167)
(133,161)(134,162)(135,164)(136,163)(137,169)(138,170)(139,172)(140,171)
(141,173)(142,174)(143,176)(144,175);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope