# Polytope of Type {78,4}

Atlas Canonical Name : {78,4}*624c
if this polytope has a name.
Group : SmallGroup(624,245)
Rank : 3
Schlafli Type : {78,4}
Number of vertices, edges, etc : 78, 156, 4
Order of s0s1s2 : 39
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{78,4,2} of size 1248
Vertex Figure Of :
{2,78,4} of size 1248
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {39,4}*312
13-fold quotients : {6,4}*48b
26-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {78,4}*1248
3-fold covers : {234,4}*1872c, {78,12}*1872d
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5, 49)(  6, 51)(  7, 50)(  8, 52)(  9, 45)( 10, 47)( 11, 46)
( 12, 48)( 13, 41)( 14, 43)( 15, 42)( 16, 44)( 17, 37)( 18, 39)( 19, 38)
( 20, 40)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 25, 29)( 26, 31)( 27, 30)
( 28, 32)( 53,105)( 54,107)( 55,106)( 56,108)( 57,153)( 58,155)( 59,154)
( 60,156)( 61,149)( 62,151)( 63,150)( 64,152)( 65,145)( 66,147)( 67,146)
( 68,148)( 69,141)( 70,143)( 71,142)( 72,144)( 73,137)( 74,139)( 75,138)
( 76,140)( 77,133)( 78,135)( 79,134)( 80,136)( 81,129)( 82,131)( 83,130)
( 84,132)( 85,125)( 86,127)( 87,126)( 88,128)( 89,121)( 90,123)( 91,122)
( 92,124)( 93,117)( 94,119)( 95,118)( 96,120)( 97,113)( 98,115)( 99,114)
(100,116)(101,109)(102,111)(103,110)(104,112)(158,159)(161,205)(162,207)
(163,206)(164,208)(165,201)(166,203)(167,202)(168,204)(169,197)(170,199)
(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,189)(178,191)
(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(209,261)(210,263)
(211,262)(212,264)(213,309)(214,311)(215,310)(216,312)(217,305)(218,307)
(219,306)(220,308)(221,301)(222,303)(223,302)(224,304)(225,297)(226,299)
(227,298)(228,300)(229,293)(230,295)(231,294)(232,296)(233,289)(234,291)
(235,290)(236,292)(237,285)(238,287)(239,286)(240,288)(241,281)(242,283)
(243,282)(244,284)(245,277)(246,279)(247,278)(248,280)(249,273)(250,275)
(251,274)(252,276)(253,269)(254,271)(255,270)(256,272)(257,265)(258,267)
(259,266)(260,268);;
s1 := (  1,213)(  2,214)(  3,216)(  4,215)(  5,209)(  6,210)(  7,212)(  8,211)
(  9,257)( 10,258)( 11,260)( 12,259)( 13,253)( 14,254)( 15,256)( 16,255)
( 17,249)( 18,250)( 19,252)( 20,251)( 21,245)( 22,246)( 23,248)( 24,247)
( 25,241)( 26,242)( 27,244)( 28,243)( 29,237)( 30,238)( 31,240)( 32,239)
( 33,233)( 34,234)( 35,236)( 36,235)( 37,229)( 38,230)( 39,232)( 40,231)
( 41,225)( 42,226)( 43,228)( 44,227)( 45,221)( 46,222)( 47,224)( 48,223)
( 49,217)( 50,218)( 51,220)( 52,219)( 53,161)( 54,162)( 55,164)( 56,163)
( 57,157)( 58,158)( 59,160)( 60,159)( 61,205)( 62,206)( 63,208)( 64,207)
( 65,201)( 66,202)( 67,204)( 68,203)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,193)( 74,194)( 75,196)( 76,195)( 77,189)( 78,190)( 79,192)( 80,191)
( 81,185)( 82,186)( 83,188)( 84,187)( 85,181)( 86,182)( 87,184)( 88,183)
( 89,177)( 90,178)( 91,180)( 92,179)( 93,173)( 94,174)( 95,176)( 96,175)
( 97,169)( 98,170)( 99,172)(100,171)(101,165)(102,166)(103,168)(104,167)
(105,265)(106,266)(107,268)(108,267)(109,261)(110,262)(111,264)(112,263)
(113,309)(114,310)(115,312)(116,311)(117,305)(118,306)(119,308)(120,307)
(121,301)(122,302)(123,304)(124,303)(125,297)(126,298)(127,300)(128,299)
(129,293)(130,294)(131,296)(132,295)(133,289)(134,290)(135,292)(136,291)
(137,285)(138,286)(139,288)(140,287)(141,281)(142,282)(143,284)(144,283)
(145,277)(146,278)(147,280)(148,279)(149,273)(150,274)(151,276)(152,275)
(153,269)(154,270)(155,272)(156,271);;
s2 := (  1,160)(  2,159)(  3,158)(  4,157)(  5,164)(  6,163)(  7,162)(  8,161)
(  9,168)( 10,167)( 11,166)( 12,165)( 13,172)( 14,171)( 15,170)( 16,169)
( 17,176)( 18,175)( 19,174)( 20,173)( 21,180)( 22,179)( 23,178)( 24,177)
( 25,184)( 26,183)( 27,182)( 28,181)( 29,188)( 30,187)( 31,186)( 32,185)
( 33,192)( 34,191)( 35,190)( 36,189)( 37,196)( 38,195)( 39,194)( 40,193)
( 41,200)( 42,199)( 43,198)( 44,197)( 45,204)( 46,203)( 47,202)( 48,201)
( 49,208)( 50,207)( 51,206)( 52,205)( 53,212)( 54,211)( 55,210)( 56,209)
( 57,216)( 58,215)( 59,214)( 60,213)( 61,220)( 62,219)( 63,218)( 64,217)
( 65,224)( 66,223)( 67,222)( 68,221)( 69,228)( 70,227)( 71,226)( 72,225)
( 73,232)( 74,231)( 75,230)( 76,229)( 77,236)( 78,235)( 79,234)( 80,233)
( 81,240)( 82,239)( 83,238)( 84,237)( 85,244)( 86,243)( 87,242)( 88,241)
( 89,248)( 90,247)( 91,246)( 92,245)( 93,252)( 94,251)( 95,250)( 96,249)
( 97,256)( 98,255)( 99,254)(100,253)(101,260)(102,259)(103,258)(104,257)
(105,264)(106,263)(107,262)(108,261)(109,268)(110,267)(111,266)(112,265)
(113,272)(114,271)(115,270)(116,269)(117,276)(118,275)(119,274)(120,273)
(121,280)(122,279)(123,278)(124,277)(125,284)(126,283)(127,282)(128,281)
(129,288)(130,287)(131,286)(132,285)(133,292)(134,291)(135,290)(136,289)
(137,296)(138,295)(139,294)(140,293)(141,300)(142,299)(143,298)(144,297)
(145,304)(146,303)(147,302)(148,301)(149,308)(150,307)(151,306)(152,305)
(153,312)(154,311)(155,310)(156,309);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(312)!(  2,  3)(  5, 49)(  6, 51)(  7, 50)(  8, 52)(  9, 45)( 10, 47)
( 11, 46)( 12, 48)( 13, 41)( 14, 43)( 15, 42)( 16, 44)( 17, 37)( 18, 39)
( 19, 38)( 20, 40)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 25, 29)( 26, 31)
( 27, 30)( 28, 32)( 53,105)( 54,107)( 55,106)( 56,108)( 57,153)( 58,155)
( 59,154)( 60,156)( 61,149)( 62,151)( 63,150)( 64,152)( 65,145)( 66,147)
( 67,146)( 68,148)( 69,141)( 70,143)( 71,142)( 72,144)( 73,137)( 74,139)
( 75,138)( 76,140)( 77,133)( 78,135)( 79,134)( 80,136)( 81,129)( 82,131)
( 83,130)( 84,132)( 85,125)( 86,127)( 87,126)( 88,128)( 89,121)( 90,123)
( 91,122)( 92,124)( 93,117)( 94,119)( 95,118)( 96,120)( 97,113)( 98,115)
( 99,114)(100,116)(101,109)(102,111)(103,110)(104,112)(158,159)(161,205)
(162,207)(163,206)(164,208)(165,201)(166,203)(167,202)(168,204)(169,197)
(170,199)(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,189)
(178,191)(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(209,261)
(210,263)(211,262)(212,264)(213,309)(214,311)(215,310)(216,312)(217,305)
(218,307)(219,306)(220,308)(221,301)(222,303)(223,302)(224,304)(225,297)
(226,299)(227,298)(228,300)(229,293)(230,295)(231,294)(232,296)(233,289)
(234,291)(235,290)(236,292)(237,285)(238,287)(239,286)(240,288)(241,281)
(242,283)(243,282)(244,284)(245,277)(246,279)(247,278)(248,280)(249,273)
(250,275)(251,274)(252,276)(253,269)(254,271)(255,270)(256,272)(257,265)
(258,267)(259,266)(260,268);
s1 := Sym(312)!(  1,213)(  2,214)(  3,216)(  4,215)(  5,209)(  6,210)(  7,212)
(  8,211)(  9,257)( 10,258)( 11,260)( 12,259)( 13,253)( 14,254)( 15,256)
( 16,255)( 17,249)( 18,250)( 19,252)( 20,251)( 21,245)( 22,246)( 23,248)
( 24,247)( 25,241)( 26,242)( 27,244)( 28,243)( 29,237)( 30,238)( 31,240)
( 32,239)( 33,233)( 34,234)( 35,236)( 36,235)( 37,229)( 38,230)( 39,232)
( 40,231)( 41,225)( 42,226)( 43,228)( 44,227)( 45,221)( 46,222)( 47,224)
( 48,223)( 49,217)( 50,218)( 51,220)( 52,219)( 53,161)( 54,162)( 55,164)
( 56,163)( 57,157)( 58,158)( 59,160)( 60,159)( 61,205)( 62,206)( 63,208)
( 64,207)( 65,201)( 66,202)( 67,204)( 68,203)( 69,197)( 70,198)( 71,200)
( 72,199)( 73,193)( 74,194)( 75,196)( 76,195)( 77,189)( 78,190)( 79,192)
( 80,191)( 81,185)( 82,186)( 83,188)( 84,187)( 85,181)( 86,182)( 87,184)
( 88,183)( 89,177)( 90,178)( 91,180)( 92,179)( 93,173)( 94,174)( 95,176)
( 96,175)( 97,169)( 98,170)( 99,172)(100,171)(101,165)(102,166)(103,168)
(104,167)(105,265)(106,266)(107,268)(108,267)(109,261)(110,262)(111,264)
(112,263)(113,309)(114,310)(115,312)(116,311)(117,305)(118,306)(119,308)
(120,307)(121,301)(122,302)(123,304)(124,303)(125,297)(126,298)(127,300)
(128,299)(129,293)(130,294)(131,296)(132,295)(133,289)(134,290)(135,292)
(136,291)(137,285)(138,286)(139,288)(140,287)(141,281)(142,282)(143,284)
(144,283)(145,277)(146,278)(147,280)(148,279)(149,273)(150,274)(151,276)
(152,275)(153,269)(154,270)(155,272)(156,271);
s2 := Sym(312)!(  1,160)(  2,159)(  3,158)(  4,157)(  5,164)(  6,163)(  7,162)
(  8,161)(  9,168)( 10,167)( 11,166)( 12,165)( 13,172)( 14,171)( 15,170)
( 16,169)( 17,176)( 18,175)( 19,174)( 20,173)( 21,180)( 22,179)( 23,178)
( 24,177)( 25,184)( 26,183)( 27,182)( 28,181)( 29,188)( 30,187)( 31,186)
( 32,185)( 33,192)( 34,191)( 35,190)( 36,189)( 37,196)( 38,195)( 39,194)
( 40,193)( 41,200)( 42,199)( 43,198)( 44,197)( 45,204)( 46,203)( 47,202)
( 48,201)( 49,208)( 50,207)( 51,206)( 52,205)( 53,212)( 54,211)( 55,210)
( 56,209)( 57,216)( 58,215)( 59,214)( 60,213)( 61,220)( 62,219)( 63,218)
( 64,217)( 65,224)( 66,223)( 67,222)( 68,221)( 69,228)( 70,227)( 71,226)
( 72,225)( 73,232)( 74,231)( 75,230)( 76,229)( 77,236)( 78,235)( 79,234)
( 80,233)( 81,240)( 82,239)( 83,238)( 84,237)( 85,244)( 86,243)( 87,242)
( 88,241)( 89,248)( 90,247)( 91,246)( 92,245)( 93,252)( 94,251)( 95,250)
( 96,249)( 97,256)( 98,255)( 99,254)(100,253)(101,260)(102,259)(103,258)
(104,257)(105,264)(106,263)(107,262)(108,261)(109,268)(110,267)(111,266)
(112,265)(113,272)(114,271)(115,270)(116,269)(117,276)(118,275)(119,274)
(120,273)(121,280)(122,279)(123,278)(124,277)(125,284)(126,283)(127,282)
(128,281)(129,288)(130,287)(131,286)(132,285)(133,292)(134,291)(135,290)
(136,289)(137,296)(138,295)(139,294)(140,293)(141,300)(142,299)(143,298)
(144,297)(145,304)(146,303)(147,302)(148,301)(149,308)(150,307)(151,306)
(152,305)(153,312)(154,311)(155,310)(156,309);
poly := sub<Sym(312)|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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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