# Polytope of Type {75,6}

Atlas Canonical Name : {75,6}*900
if this polytope has a name.
Group : SmallGroup(900,37)
Rank : 3
Schlafli Type : {75,6}
Number of vertices, edges, etc : 75, 225, 6
Order of s0s1s2 : 150
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{75,6,2} of size 1800
Vertex Figure Of :
{2,75,6} of size 1800
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {75,2}*300
5-fold quotients : {15,6}*180
9-fold quotients : {25,2}*100
15-fold quotients : {15,2}*60
25-fold quotients : {3,6}*36
45-fold quotients : {5,2}*20
75-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {150,6}*1800c
Permutation Representation (GAP) :
```s0 := (  2,  5)(  3,  4)(  6, 25)(  7, 24)(  8, 23)(  9, 22)( 10, 21)( 11, 20)
( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 26, 51)( 27, 55)( 28, 54)( 29, 53)
( 30, 52)( 31, 75)( 32, 74)( 33, 73)( 34, 72)( 35, 71)( 36, 70)( 37, 69)
( 38, 68)( 39, 67)( 40, 66)( 41, 65)( 42, 64)( 43, 63)( 44, 62)( 45, 61)
( 46, 60)( 47, 59)( 48, 58)( 49, 57)( 50, 56)( 76,151)( 77,155)( 78,154)
( 79,153)( 80,152)( 81,175)( 82,174)( 83,173)( 84,172)( 85,171)( 86,170)
( 87,169)( 88,168)( 89,167)( 90,166)( 91,165)( 92,164)( 93,163)( 94,162)
( 95,161)( 96,160)( 97,159)( 98,158)( 99,157)(100,156)(101,201)(102,205)
(103,204)(104,203)(105,202)(106,225)(107,224)(108,223)(109,222)(110,221)
(111,220)(112,219)(113,218)(114,217)(115,216)(116,215)(117,214)(118,213)
(119,212)(120,211)(121,210)(122,209)(123,208)(124,207)(125,206)(126,176)
(127,180)(128,179)(129,178)(130,177)(131,200)(132,199)(133,198)(134,197)
(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)(141,190)(142,189)
(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)(150,181);;
s1 := (  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,101)(  7,105)(  8,104)
(  9,103)( 10,102)( 11,125)( 12,124)( 13,123)( 14,122)( 15,121)( 16,120)
( 17,119)( 18,118)( 19,117)( 20,116)( 21,115)( 22,114)( 23,113)( 24,112)
( 25,111)( 26, 81)( 27, 85)( 28, 84)( 29, 83)( 30, 82)( 31, 76)( 32, 80)
( 33, 79)( 34, 78)( 35, 77)( 36,100)( 37, 99)( 38, 98)( 39, 97)( 40, 96)
( 41, 95)( 42, 94)( 43, 93)( 44, 92)( 45, 91)( 46, 90)( 47, 89)( 48, 88)
( 49, 87)( 50, 86)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)( 56,126)
( 57,130)( 58,129)( 59,128)( 60,127)( 61,150)( 62,149)( 63,148)( 64,147)
( 65,146)( 66,145)( 67,144)( 68,143)( 69,142)( 70,141)( 71,140)( 72,139)
( 73,138)( 74,137)( 75,136)(151,181)(152,185)(153,184)(154,183)(155,182)
(156,176)(157,180)(158,179)(159,178)(160,177)(161,200)(162,199)(163,198)
(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)(171,190)
(172,189)(173,188)(174,187)(175,186)(201,206)(202,210)(203,209)(204,208)
(205,207)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219);;
s2 := ( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)( 81,156)( 82,157)( 83,158)
( 84,159)( 85,160)( 86,161)( 87,162)( 88,163)( 89,164)( 90,165)( 91,166)
( 92,167)( 93,168)( 94,169)( 95,170)( 96,171)( 97,172)( 98,173)( 99,174)
(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)
(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)
(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,197)(123,198)
(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)
(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)
(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)
(148,223)(149,224)(150,225);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(225)!(  2,  5)(  3,  4)(  6, 25)(  7, 24)(  8, 23)(  9, 22)( 10, 21)
( 11, 20)( 12, 19)( 13, 18)( 14, 17)( 15, 16)( 26, 51)( 27, 55)( 28, 54)
( 29, 53)( 30, 52)( 31, 75)( 32, 74)( 33, 73)( 34, 72)( 35, 71)( 36, 70)
( 37, 69)( 38, 68)( 39, 67)( 40, 66)( 41, 65)( 42, 64)( 43, 63)( 44, 62)
( 45, 61)( 46, 60)( 47, 59)( 48, 58)( 49, 57)( 50, 56)( 76,151)( 77,155)
( 78,154)( 79,153)( 80,152)( 81,175)( 82,174)( 83,173)( 84,172)( 85,171)
( 86,170)( 87,169)( 88,168)( 89,167)( 90,166)( 91,165)( 92,164)( 93,163)
( 94,162)( 95,161)( 96,160)( 97,159)( 98,158)( 99,157)(100,156)(101,201)
(102,205)(103,204)(104,203)(105,202)(106,225)(107,224)(108,223)(109,222)
(110,221)(111,220)(112,219)(113,218)(114,217)(115,216)(116,215)(117,214)
(118,213)(119,212)(120,211)(121,210)(122,209)(123,208)(124,207)(125,206)
(126,176)(127,180)(128,179)(129,178)(130,177)(131,200)(132,199)(133,198)
(134,197)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)(141,190)
(142,189)(143,188)(144,187)(145,186)(146,185)(147,184)(148,183)(149,182)
(150,181);
s1 := Sym(225)!(  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,101)(  7,105)
(  8,104)(  9,103)( 10,102)( 11,125)( 12,124)( 13,123)( 14,122)( 15,121)
( 16,120)( 17,119)( 18,118)( 19,117)( 20,116)( 21,115)( 22,114)( 23,113)
( 24,112)( 25,111)( 26, 81)( 27, 85)( 28, 84)( 29, 83)( 30, 82)( 31, 76)
( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36,100)( 37, 99)( 38, 98)( 39, 97)
( 40, 96)( 41, 95)( 42, 94)( 43, 93)( 44, 92)( 45, 91)( 46, 90)( 47, 89)
( 48, 88)( 49, 87)( 50, 86)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)
( 56,126)( 57,130)( 58,129)( 59,128)( 60,127)( 61,150)( 62,149)( 63,148)
( 64,147)( 65,146)( 66,145)( 67,144)( 68,143)( 69,142)( 70,141)( 71,140)
( 72,139)( 73,138)( 74,137)( 75,136)(151,181)(152,185)(153,184)(154,183)
(155,182)(156,176)(157,180)(158,179)(159,178)(160,177)(161,200)(162,199)
(163,198)(164,197)(165,196)(166,195)(167,194)(168,193)(169,192)(170,191)
(171,190)(172,189)(173,188)(174,187)(175,186)(201,206)(202,210)(203,209)
(204,208)(205,207)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)
(217,219);
s2 := Sym(225)!( 76,151)( 77,152)( 78,153)( 79,154)( 80,155)( 81,156)( 82,157)
( 83,158)( 84,159)( 85,160)( 86,161)( 87,162)( 88,163)( 89,164)( 90,165)
( 91,166)( 92,167)( 93,168)( 94,169)( 95,170)( 96,171)( 97,172)( 98,173)
( 99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)
(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)
(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,197)
(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)
(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)
(139,214)(140,215)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)
(147,222)(148,223)(149,224)(150,225);
poly := sub<Sym(225)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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