# Polytope of Type {68,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {68,6}*816a
Also Known As : {68,6|2}. if this polytope has another name.
Group : SmallGroup(816,133)
Rank : 3
Schlafli Type : {68,6}
Number of vertices, edges, etc : 68, 204, 6
Order of s0s1s2 : 204
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{68,6,2} of size 1632
Vertex Figure Of :
{2,68,6} of size 1632
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {34,6}*408
3-fold quotients : {68,2}*272
6-fold quotients : {34,2}*136
12-fold quotients : {17,2}*68
17-fold quotients : {4,6}*48a
34-fold quotients : {2,6}*24
51-fold quotients : {4,2}*16
68-fold quotients : {2,3}*12
102-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {136,6}*1632, {68,12}*1632
Permutation Representation (GAP) :
```s0 := (  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)
( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 25, 28)( 26, 27)
( 36, 51)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)( 43, 44)
( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)( 60, 61)
( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)( 77, 78)
( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)
(103,154)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)
(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)
(119,155)(120,171)(121,187)(122,186)(123,185)(124,184)(125,183)(126,182)
(127,181)(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)
(135,173)(136,172)(137,188)(138,204)(139,203)(140,202)(141,201)(142,200)
(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)
(151,191)(152,190)(153,189);;
s1 := (  1,104)(  2,103)(  3,119)(  4,118)(  5,117)(  6,116)(  7,115)(  8,114)
(  9,113)( 10,112)( 11,111)( 12,110)( 13,109)( 14,108)( 15,107)( 16,106)
( 17,105)( 18,138)( 19,137)( 20,153)( 21,152)( 22,151)( 23,150)( 24,149)
( 25,148)( 26,147)( 27,146)( 28,145)( 29,144)( 30,143)( 31,142)( 32,141)
( 33,140)( 34,139)( 35,121)( 36,120)( 37,136)( 38,135)( 39,134)( 40,133)
( 41,132)( 42,131)( 43,130)( 44,129)( 45,128)( 46,127)( 47,126)( 48,125)
( 49,124)( 50,123)( 51,122)( 52,155)( 53,154)( 54,170)( 55,169)( 56,168)
( 57,167)( 58,166)( 59,165)( 60,164)( 61,163)( 62,162)( 63,161)( 64,160)
( 65,159)( 66,158)( 67,157)( 68,156)( 69,189)( 70,188)( 71,204)( 72,203)
( 73,202)( 74,201)( 75,200)( 76,199)( 77,198)( 78,197)( 79,196)( 80,195)
( 81,194)( 82,193)( 83,192)( 84,191)( 85,190)( 86,172)( 87,171)( 88,187)
( 89,186)( 90,185)( 91,184)( 92,183)( 93,182)( 94,181)( 95,180)( 96,179)
( 97,178)( 98,177)( 99,176)(100,175)(101,174)(102,173);;
s2 := (  1, 18)(  2, 19)(  3, 20)(  4, 21)(  5, 22)(  6, 23)(  7, 24)(  8, 25)
(  9, 26)( 10, 27)( 11, 28)( 12, 29)( 13, 30)( 14, 31)( 15, 32)( 16, 33)
( 17, 34)( 52, 69)( 53, 70)( 54, 71)( 55, 72)( 56, 73)( 57, 74)( 58, 75)
( 59, 76)( 60, 77)( 61, 78)( 62, 79)( 63, 80)( 64, 81)( 65, 82)( 66, 83)
( 67, 84)( 68, 85)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)
(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)
(117,134)(118,135)(119,136)(154,171)(155,172)(156,173)(157,174)(158,175)
(159,176)(160,177)(161,178)(162,179)(163,180)(164,181)(165,182)(166,183)
(167,184)(168,185)(169,186)(170,187);;
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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*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*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*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*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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(204)!(  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)
(  9, 10)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 25, 28)
( 26, 27)( 36, 51)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 43, 44)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)
( 60, 61)( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)
( 77, 78)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)
( 94, 95)(103,154)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)
(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)
(118,156)(119,155)(120,171)(121,187)(122,186)(123,185)(124,184)(125,183)
(126,182)(127,181)(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)
(134,174)(135,173)(136,172)(137,188)(138,204)(139,203)(140,202)(141,201)
(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)
(150,192)(151,191)(152,190)(153,189);
s1 := Sym(204)!(  1,104)(  2,103)(  3,119)(  4,118)(  5,117)(  6,116)(  7,115)
(  8,114)(  9,113)( 10,112)( 11,111)( 12,110)( 13,109)( 14,108)( 15,107)
( 16,106)( 17,105)( 18,138)( 19,137)( 20,153)( 21,152)( 22,151)( 23,150)
( 24,149)( 25,148)( 26,147)( 27,146)( 28,145)( 29,144)( 30,143)( 31,142)
( 32,141)( 33,140)( 34,139)( 35,121)( 36,120)( 37,136)( 38,135)( 39,134)
( 40,133)( 41,132)( 42,131)( 43,130)( 44,129)( 45,128)( 46,127)( 47,126)
( 48,125)( 49,124)( 50,123)( 51,122)( 52,155)( 53,154)( 54,170)( 55,169)
( 56,168)( 57,167)( 58,166)( 59,165)( 60,164)( 61,163)( 62,162)( 63,161)
( 64,160)( 65,159)( 66,158)( 67,157)( 68,156)( 69,189)( 70,188)( 71,204)
( 72,203)( 73,202)( 74,201)( 75,200)( 76,199)( 77,198)( 78,197)( 79,196)
( 80,195)( 81,194)( 82,193)( 83,192)( 84,191)( 85,190)( 86,172)( 87,171)
( 88,187)( 89,186)( 90,185)( 91,184)( 92,183)( 93,182)( 94,181)( 95,180)
( 96,179)( 97,178)( 98,177)( 99,176)(100,175)(101,174)(102,173);
s2 := Sym(204)!(  1, 18)(  2, 19)(  3, 20)(  4, 21)(  5, 22)(  6, 23)(  7, 24)
(  8, 25)(  9, 26)( 10, 27)( 11, 28)( 12, 29)( 13, 30)( 14, 31)( 15, 32)
( 16, 33)( 17, 34)( 52, 69)( 53, 70)( 54, 71)( 55, 72)( 56, 73)( 57, 74)
( 58, 75)( 59, 76)( 60, 77)( 61, 78)( 62, 79)( 63, 80)( 64, 81)( 65, 82)
( 66, 83)( 67, 84)( 68, 85)(103,120)(104,121)(105,122)(106,123)(107,124)
(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)
(116,133)(117,134)(118,135)(119,136)(154,171)(155,172)(156,173)(157,174)
(158,175)(159,176)(160,177)(161,178)(162,179)(163,180)(164,181)(165,182)
(166,183)(167,184)(168,185)(169,186)(170,187);
poly := sub<Sym(204)|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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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*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*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*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*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*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope