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# Polytope of Type {34,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {34,10}*680
Also Known As : {34,10|2}. if this polytope has another name.
Group : SmallGroup(680,49)
Rank : 3
Schlafli Type : {34,10}
Number of vertices, edges, etc : 34, 170, 10
Order of s0s1s2 : 170
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{34,10,2} of size 1360
Vertex Figure Of :
{2,34,10} of size 1360
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {34,2}*136
10-fold quotients : {17,2}*68
17-fold quotients : {2,10}*40
34-fold quotients : {2,5}*20
85-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {34,20}*1360, {68,10}*1360
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)
(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)
(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)
(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146)
(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163);;
s1 := (  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)
( 18, 70)( 19, 69)( 20, 85)( 21, 84)( 22, 83)( 23, 82)( 24, 81)( 25, 80)
( 26, 79)( 27, 78)( 28, 77)( 29, 76)( 30, 75)( 31, 74)( 32, 73)( 33, 72)
( 34, 71)( 35, 53)( 36, 52)( 37, 68)( 38, 67)( 39, 66)( 40, 65)( 41, 64)
( 42, 63)( 43, 62)( 44, 61)( 45, 60)( 46, 59)( 47, 58)( 48, 57)( 49, 56)
( 50, 55)( 51, 54)( 86, 87)( 88,102)( 89,101)( 90,100)( 91, 99)( 92, 98)
( 93, 97)( 94, 96)(103,155)(104,154)(105,170)(106,169)(107,168)(108,167)
(109,166)(110,165)(111,164)(112,163)(113,162)(114,161)(115,160)(116,159)
(117,158)(118,157)(119,156)(120,138)(121,137)(122,153)(123,152)(124,151)
(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)
(133,142)(134,141)(135,140)(136,139);;
s2 := (  1,103)(  2,104)(  3,105)(  4,106)(  5,107)(  6,108)(  7,109)(  8,110)
(  9,111)( 10,112)( 11,113)( 12,114)( 13,115)( 14,116)( 15,117)( 16,118)
( 17,119)( 18, 86)( 19, 87)( 20, 88)( 21, 89)( 22, 90)( 23, 91)( 24, 92)
( 25, 93)( 26, 94)( 27, 95)( 28, 96)( 29, 97)( 30, 98)( 31, 99)( 32,100)
( 33,101)( 34,102)( 35,154)( 36,155)( 37,156)( 38,157)( 39,158)( 40,159)
( 41,160)( 42,161)( 43,162)( 44,163)( 45,164)( 46,165)( 47,166)( 48,167)
( 49,168)( 50,169)( 51,170)( 52,137)( 53,138)( 54,139)( 55,140)( 56,141)
( 57,142)( 58,143)( 59,144)( 60,145)( 61,146)( 62,147)( 63,148)( 64,149)
( 65,150)( 66,151)( 67,152)( 68,153)( 69,120)( 70,121)( 71,122)( 72,123)
( 73,124)( 74,125)( 75,126)( 76,127)( 77,128)( 78,129)( 79,130)( 80,131)
( 81,132)( 82,133)( 83,134)( 84,135)( 85,136);;
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*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope