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# Polytope of Type {16,6}

Atlas Canonical Name : {16,6}*768c
if this polytope has a name.
Group : SmallGroup(768,1088591)
Rank : 3
Schlafli Type : {16,6}
Number of vertices, edges, etc : 64, 192, 24
Order of s0s1s2 : 48
Order of s0s1s2s1 : 8
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 : {8,6}*384g
4-fold quotients : {4,6}*192b
8-fold quotients : {8,6}*96, {4,6}*96
16-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {8,2}*32
32-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {4,2}*16
64-fold quotients : {2,3}*12
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,103)(  2,104)(  3,101)(  4,102)(  5,100)(  6, 99)(  7, 98)(  8, 97)
(  9,111)( 10,112)( 11,109)( 12,110)( 13,108)( 14,107)( 15,106)( 16,105)
( 17,119)( 18,120)( 19,117)( 20,118)( 21,116)( 22,115)( 23,114)( 24,113)
( 25,128)( 26,127)( 27,126)( 28,125)( 29,123)( 30,124)( 31,121)( 32,122)
( 33,136)( 34,135)( 35,134)( 36,133)( 37,131)( 38,132)( 39,129)( 40,130)
( 41,144)( 42,143)( 43,142)( 44,141)( 45,139)( 46,140)( 47,137)( 48,138)
( 49,175)( 50,176)( 51,173)( 52,174)( 53,172)( 54,171)( 55,170)( 56,169)
( 57,183)( 58,184)( 59,181)( 60,182)( 61,180)( 62,179)( 63,178)( 64,177)
( 65,191)( 66,192)( 67,189)( 68,190)( 69,188)( 70,187)( 71,186)( 72,185)
( 73,151)( 74,152)( 75,149)( 76,150)( 77,148)( 78,147)( 79,146)( 80,145)
( 81,159)( 82,160)( 83,157)( 84,158)( 85,156)( 86,155)( 87,154)( 88,153)
( 89,167)( 90,168)( 91,165)( 92,166)( 93,164)( 94,163)( 95,162)( 96,161);;
s1 := (  3,  4)(  5,  7)(  6,  8)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 25, 26)( 29, 32)( 30, 31)( 33, 42)( 34, 41)
( 35, 43)( 36, 44)( 37, 48)( 38, 47)( 39, 46)( 40, 45)( 49, 73)( 50, 74)
( 51, 76)( 52, 75)( 53, 79)( 54, 80)( 55, 77)( 56, 78)( 57, 89)( 58, 90)
( 59, 92)( 60, 91)( 61, 95)( 62, 96)( 63, 93)( 64, 94)( 65, 81)( 66, 82)
( 67, 84)( 68, 83)( 69, 87)( 70, 88)( 71, 85)( 72, 86)( 97,146)( 98,145)
( 99,147)(100,148)(101,152)(102,151)(103,150)(104,149)(105,162)(106,161)
(107,163)(108,164)(109,168)(110,167)(111,166)(112,165)(113,154)(114,153)
(115,155)(116,156)(117,160)(118,159)(119,158)(120,157)(121,169)(122,170)
(123,172)(124,171)(125,175)(126,176)(127,173)(128,174)(129,185)(130,186)
(131,188)(132,187)(133,191)(134,192)(135,189)(136,190)(137,177)(138,178)
(139,180)(140,179)(141,183)(142,184)(143,181)(144,182);;
s2 := (  1, 17)(  2, 18)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 24)(  8, 23)
( 11, 14)( 12, 13)( 15, 16)( 25, 41)( 26, 42)( 27, 46)( 28, 45)( 29, 44)
( 30, 43)( 31, 48)( 32, 47)( 35, 38)( 36, 37)( 39, 40)( 49, 65)( 50, 66)
( 51, 70)( 52, 69)( 53, 68)( 54, 67)( 55, 72)( 56, 71)( 59, 62)( 60, 61)
( 63, 64)( 73, 89)( 74, 90)( 75, 94)( 76, 93)( 77, 92)( 78, 91)( 79, 96)
( 80, 95)( 83, 86)( 84, 85)( 87, 88)( 97,114)( 98,113)( 99,117)(100,118)
(101,115)(102,116)(103,119)(104,120)(105,106)(107,109)(108,110)(121,138)
(122,137)(123,141)(124,142)(125,139)(126,140)(127,143)(128,144)(129,130)
(131,133)(132,134)(145,162)(146,161)(147,165)(148,166)(149,163)(150,164)
(151,167)(152,168)(153,154)(155,157)(156,158)(169,186)(170,185)(171,189)
(172,190)(173,187)(174,188)(175,191)(176,192)(177,178)(179,181)(180,182);;
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*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope