# Polytope of Type {78,6}

Atlas Canonical Name : {78,6}*936b
Also Known As : {78,6|2}. if this polytope has another name.
Group : SmallGroup(936,212)
Rank : 3
Schlafli Type : {78,6}
Number of vertices, edges, etc : 78, 234, 6
Order of s0s1s2 : 78
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{78,6,2} of size 1872
Vertex Figure Of :
{2,78,6} of size 1872
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {26,6}*312, {78,2}*312
6-fold quotients : {39,2}*156
9-fold quotients : {26,2}*104
13-fold quotients : {6,6}*72a
18-fold quotients : {13,2}*52
39-fold quotients : {2,6}*24, {6,2}*24
78-fold quotients : {2,3}*12, {3,2}*12
117-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {78,12}*1872b, {156,6}*1872b
Permutation Representation (GAP) :
```s0 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)( 15, 39)
( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)
( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 80, 91)( 81, 90)
( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)( 95,115)
( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)(103,107)
(104,106)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(131,144)
(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)
(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)(161,166)
(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)(175,191)
(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(197,208)
(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)(210,234)(211,233)
(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)(218,226)(219,225)
(220,224)(221,223);;
s1 := (  1, 15)(  2, 14)(  3, 26)(  4, 25)(  5, 24)(  6, 23)(  7, 22)(  8, 21)
(  9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)( 44,102)
( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)( 52, 94)
( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)( 60, 86)
( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)( 68,117)
( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,108)( 78,107)(118,132)(119,131)(120,143)(121,142)(122,141)(123,140)
(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(144,145)
(146,156)(147,155)(148,154)(149,153)(150,152)(157,210)(158,209)(159,221)
(160,220)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)(167,213)
(168,212)(169,211)(170,197)(171,196)(172,208)(173,207)(174,206)(175,205)
(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)(183,223)
(184,222)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)(191,228)
(192,227)(193,226)(194,225)(195,224);;
s2 := (  1,157)(  2,158)(  3,159)(  4,160)(  5,161)(  6,162)(  7,163)(  8,164)
(  9,165)( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)( 16,172)
( 17,173)( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)( 24,180)
( 25,181)( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)( 32,188)
( 33,189)( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)( 40,118)
( 41,119)( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)( 48,126)
( 49,127)( 50,128)( 51,129)( 52,130)( 53,131)( 54,132)( 55,133)( 56,134)
( 57,135)( 58,136)( 59,137)( 60,138)( 61,139)( 62,140)( 63,141)( 64,142)
( 65,143)( 66,144)( 67,145)( 68,146)( 69,147)( 70,148)( 71,149)( 72,150)
( 73,151)( 74,152)( 75,153)( 76,154)( 77,155)( 78,156)( 79,196)( 80,197)
( 81,198)( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)( 88,205)
( 89,206)( 90,207)( 91,208)( 92,209)( 93,210)( 94,211)( 95,212)( 96,213)
( 97,214)( 98,215)( 99,216)(100,217)(101,218)(102,219)(103,220)(104,221)
(105,222)(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)(112,229)
(113,230)(114,231)(115,232)(116,233)(117,234);;
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*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(234)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)
( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)
( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)
( 45, 48)( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 80, 91)
( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)
( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)
(103,107)(104,106)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)
(131,144)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)
(139,149)(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)
(161,166)(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)
(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)
(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)(210,234)
(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)(218,226)
(219,225)(220,224)(221,223);
s1 := Sym(234)!(  1, 15)(  2, 14)(  3, 26)(  4, 25)(  5, 24)(  6, 23)(  7, 22)
(  8, 21)(  9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)
( 30, 38)( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)
( 44,102)( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)
( 52, 94)( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)
( 60, 86)( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)
( 68,117)( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)
( 76,109)( 77,108)( 78,107)(118,132)(119,131)(120,143)(121,142)(122,141)
(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)
(144,145)(146,156)(147,155)(148,154)(149,153)(150,152)(157,210)(158,209)
(159,221)(160,220)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)
(167,213)(168,212)(169,211)(170,197)(171,196)(172,208)(173,207)(174,206)
(175,205)(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)
(183,223)(184,222)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)
(191,228)(192,227)(193,226)(194,225)(195,224);
s2 := Sym(234)!(  1,157)(  2,158)(  3,159)(  4,160)(  5,161)(  6,162)(  7,163)
(  8,164)(  9,165)( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)
( 16,172)( 17,173)( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)
( 24,180)( 25,181)( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)
( 32,188)( 33,189)( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)
( 40,118)( 41,119)( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)
( 48,126)( 49,127)( 50,128)( 51,129)( 52,130)( 53,131)( 54,132)( 55,133)
( 56,134)( 57,135)( 58,136)( 59,137)( 60,138)( 61,139)( 62,140)( 63,141)
( 64,142)( 65,143)( 66,144)( 67,145)( 68,146)( 69,147)( 70,148)( 71,149)
( 72,150)( 73,151)( 74,152)( 75,153)( 76,154)( 77,155)( 78,156)( 79,196)
( 80,197)( 81,198)( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)
( 88,205)( 89,206)( 90,207)( 91,208)( 92,209)( 93,210)( 94,211)( 95,212)
( 96,213)( 97,214)( 98,215)( 99,216)(100,217)(101,218)(102,219)(103,220)
(104,221)(105,222)(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)
(112,229)(113,230)(114,231)(115,232)(116,233)(117,234);
poly := sub<Sym(234)|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*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