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# Polytope of Type {8,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*384h
if this polytope has a name.
Group : SmallGroup(384,17986)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 16, 96, 24
Order of s0s1s2 : 12
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{8,12,2} of size 768
Vertex Figure Of :
{2,8,12} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12}*192c, {8,6}*192b
4-fold quotients : {8,3}*96, {4,6}*96
8-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {4,3}*24, {2,6}*24
32-fold quotients : {2,3}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,24}*768n, {8,24}*768p, {8,12}*768s
3-fold covers : {8,36}*1152h, {24,12}*1152m, {24,12}*1152v
5-fold covers : {40,12}*1920h, {8,60}*1920h
Permutation Representation (GAP) :
```s0 := (  1, 53)(  2, 54)(  3, 56)(  4, 55)(  5, 50)(  6, 49)(  7, 51)(  8, 52)
(  9, 61)( 10, 62)( 11, 64)( 12, 63)( 13, 58)( 14, 57)( 15, 59)( 16, 60)
( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 66)( 22, 65)( 23, 67)( 24, 68)
( 25, 77)( 26, 78)( 27, 80)( 28, 79)( 29, 74)( 30, 73)( 31, 75)( 32, 76)
( 33, 85)( 34, 86)( 35, 88)( 36, 87)( 37, 82)( 38, 81)( 39, 83)( 40, 84)
( 41, 93)( 42, 94)( 43, 96)( 44, 95)( 45, 90)( 46, 89)( 47, 91)( 48, 92)
( 97,149)( 98,150)( 99,152)(100,151)(101,146)(102,145)(103,147)(104,148)
(105,157)(106,158)(107,160)(108,159)(109,154)(110,153)(111,155)(112,156)
(113,165)(114,166)(115,168)(116,167)(117,162)(118,161)(119,163)(120,164)
(121,173)(122,174)(123,176)(124,175)(125,170)(126,169)(127,171)(128,172)
(129,181)(130,182)(131,184)(132,183)(133,178)(134,177)(135,179)(136,180)
(137,189)(138,190)(139,192)(140,191)(141,186)(142,185)(143,187)(144,188);;
s1 := (  1,145)(  2,146)(  3,150)(  4,149)(  5,148)(  6,147)(  7,152)(  8,151)
(  9,161)( 10,162)( 11,166)( 12,165)( 13,164)( 14,163)( 15,168)( 16,167)
( 17,153)( 18,154)( 19,158)( 20,157)( 21,156)( 22,155)( 23,160)( 24,159)
( 25,169)( 26,170)( 27,174)( 28,173)( 29,172)( 30,171)( 31,176)( 32,175)
( 33,185)( 34,186)( 35,190)( 36,189)( 37,188)( 38,187)( 39,192)( 40,191)
( 41,177)( 42,178)( 43,182)( 44,181)( 45,180)( 46,179)( 47,184)( 48,183)
( 49,121)( 50,122)( 51,126)( 52,125)( 53,124)( 54,123)( 55,128)( 56,127)
( 57,137)( 58,138)( 59,142)( 60,141)( 61,140)( 62,139)( 63,144)( 64,143)
( 65,129)( 66,130)( 67,134)( 68,133)( 69,132)( 70,131)( 71,136)( 72,135)
( 73, 97)( 74, 98)( 75,102)( 76,101)( 77,100)( 78, 99)( 79,104)( 80,103)
( 81,113)( 82,114)( 83,118)( 84,117)( 85,116)( 86,115)( 87,120)( 88,119)
( 89,105)( 90,106)( 91,110)( 92,109)( 93,108)( 94,107)( 95,112)( 96,111);;
s2 := (  1, 65)(  2, 66)(  3, 71)(  4, 72)(  5, 70)(  6, 69)(  7, 67)(  8, 68)
(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 62)( 14, 61)( 15, 59)( 16, 60)
( 17, 49)( 18, 50)( 19, 55)( 20, 56)( 21, 54)( 22, 53)( 23, 51)( 24, 52)
( 25, 89)( 26, 90)( 27, 95)( 28, 96)( 29, 94)( 30, 93)( 31, 91)( 32, 92)
( 33, 81)( 34, 82)( 35, 87)( 36, 88)( 37, 86)( 38, 85)( 39, 83)( 40, 84)
( 41, 73)( 42, 74)( 43, 79)( 44, 80)( 45, 78)( 46, 77)( 47, 75)( 48, 76)
( 97,161)( 98,162)( 99,167)(100,168)(101,166)(102,165)(103,163)(104,164)
(105,153)(106,154)(107,159)(108,160)(109,158)(110,157)(111,155)(112,156)
(113,145)(114,146)(115,151)(116,152)(117,150)(118,149)(119,147)(120,148)
(121,185)(122,186)(123,191)(124,192)(125,190)(126,189)(127,187)(128,188)
(129,177)(130,178)(131,183)(132,184)(133,182)(134,181)(135,179)(136,180)
(137,169)(138,170)(139,175)(140,176)(141,174)(142,173)(143,171)(144,172);;
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*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope