Polytope of Type {4,177}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,177}*1416
if this polytope has a name.
Group : SmallGroup(1416,32)
Rank : 3
Schlafli Type : {4,177}
Number of vertices, edges, etc : 4, 354, 177
Order of s₀s₁s₂ : 177
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   59-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236);;
s1 := (  3,  4)(  5,233)(  6,234)(  7,236)(  8,235)(  9,229)( 10,230)( 11,232)
( 12,231)( 13,225)( 14,226)( 15,228)( 16,227)( 17,221)( 18,222)( 19,224)
( 20,223)( 21,217)( 22,218)( 23,220)( 24,219)( 25,213)( 26,214)( 27,216)
( 28,215)( 29,209)( 30,210)( 31,212)( 32,211)( 33,205)( 34,206)( 35,208)
( 36,207)( 37,201)( 38,202)( 39,204)( 40,203)( 41,197)( 42,198)( 43,200)
( 44,199)( 45,193)( 46,194)( 47,196)( 48,195)( 49,189)( 50,190)( 51,192)
( 52,191)( 53,185)( 54,186)( 55,188)( 56,187)( 57,181)( 58,182)( 59,184)
( 60,183)( 61,177)( 62,178)( 63,180)( 64,179)( 65,173)( 66,174)( 67,176)
( 68,175)( 69,169)( 70,170)( 71,172)( 72,171)( 73,165)( 74,166)( 75,168)
( 76,167)( 77,161)( 78,162)( 79,164)( 80,163)( 81,157)( 82,158)( 83,160)
( 84,159)( 85,153)( 86,154)( 87,156)( 88,155)( 89,149)( 90,150)( 91,152)
( 92,151)( 93,145)( 94,146)( 95,148)( 96,147)( 97,141)( 98,142)( 99,144)
(100,143)(101,137)(102,138)(103,140)(104,139)(105,133)(106,134)(107,136)
(108,135)(109,129)(110,130)(111,132)(112,131)(113,125)(114,126)(115,128)
(116,127)(117,121)(118,122)(119,124)(120,123);;
s2 := (  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9,233)( 10,236)( 11,235)( 12,234)
( 13,229)( 14,232)( 15,231)( 16,230)( 17,225)( 18,228)( 19,227)( 20,226)
( 21,221)( 22,224)( 23,223)( 24,222)( 25,217)( 26,220)( 27,219)( 28,218)
( 29,213)( 30,216)( 31,215)( 32,214)( 33,209)( 34,212)( 35,211)( 36,210)
( 37,205)( 38,208)( 39,207)( 40,206)( 41,201)( 42,204)( 43,203)( 44,202)
( 45,197)( 46,200)( 47,199)( 48,198)( 49,193)( 50,196)( 51,195)( 52,194)
( 53,189)( 54,192)( 55,191)( 56,190)( 57,185)( 58,188)( 59,187)( 60,186)
( 61,181)( 62,184)( 63,183)( 64,182)( 65,177)( 66,180)( 67,179)( 68,178)
( 69,173)( 70,176)( 71,175)( 72,174)( 73,169)( 74,172)( 75,171)( 76,170)
( 77,165)( 78,168)( 79,167)( 80,166)( 81,161)( 82,164)( 83,163)( 84,162)
( 85,157)( 86,160)( 87,159)( 88,158)( 89,153)( 90,156)( 91,155)( 92,154)
( 93,149)( 94,152)( 95,151)( 96,150)( 97,145)( 98,148)( 99,147)(100,146)
(101,141)(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)(108,138)
(109,133)(110,136)(111,135)(112,134)(113,129)(114,132)(115,131)(116,130)
(117,125)(118,128)(119,127)(120,126)(122,124);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(236)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236);
s1 := Sym(236)!(  3,  4)(  5,233)(  6,234)(  7,236)(  8,235)(  9,229)( 10,230)
( 11,232)( 12,231)( 13,225)( 14,226)( 15,228)( 16,227)( 17,221)( 18,222)
( 19,224)( 20,223)( 21,217)( 22,218)( 23,220)( 24,219)( 25,213)( 26,214)
( 27,216)( 28,215)( 29,209)( 30,210)( 31,212)( 32,211)( 33,205)( 34,206)
( 35,208)( 36,207)( 37,201)( 38,202)( 39,204)( 40,203)( 41,197)( 42,198)
( 43,200)( 44,199)( 45,193)( 46,194)( 47,196)( 48,195)( 49,189)( 50,190)
( 51,192)( 52,191)( 53,185)( 54,186)( 55,188)( 56,187)( 57,181)( 58,182)
( 59,184)( 60,183)( 61,177)( 62,178)( 63,180)( 64,179)( 65,173)( 66,174)
( 67,176)( 68,175)( 69,169)( 70,170)( 71,172)( 72,171)( 73,165)( 74,166)
( 75,168)( 76,167)( 77,161)( 78,162)( 79,164)( 80,163)( 81,157)( 82,158)
( 83,160)( 84,159)( 85,153)( 86,154)( 87,156)( 88,155)( 89,149)( 90,150)
( 91,152)( 92,151)( 93,145)( 94,146)( 95,148)( 96,147)( 97,141)( 98,142)
( 99,144)(100,143)(101,137)(102,138)(103,140)(104,139)(105,133)(106,134)
(107,136)(108,135)(109,129)(110,130)(111,132)(112,131)(113,125)(114,126)
(115,128)(116,127)(117,121)(118,122)(119,124)(120,123);
s2 := Sym(236)!(  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9,233)( 10,236)( 11,235)
( 12,234)( 13,229)( 14,232)( 15,231)( 16,230)( 17,225)( 18,228)( 19,227)
( 20,226)( 21,221)( 22,224)( 23,223)( 24,222)( 25,217)( 26,220)( 27,219)
( 28,218)( 29,213)( 30,216)( 31,215)( 32,214)( 33,209)( 34,212)( 35,211)
( 36,210)( 37,205)( 38,208)( 39,207)( 40,206)( 41,201)( 42,204)( 43,203)
( 44,202)( 45,197)( 46,200)( 47,199)( 48,198)( 49,193)( 50,196)( 51,195)
( 52,194)( 53,189)( 54,192)( 55,191)( 56,190)( 57,185)( 58,188)( 59,187)
( 60,186)( 61,181)( 62,184)( 63,183)( 64,182)( 65,177)( 66,180)( 67,179)
( 68,178)( 69,173)( 70,176)( 71,175)( 72,174)( 73,169)( 74,172)( 75,171)
( 76,170)( 77,165)( 78,168)( 79,167)( 80,166)( 81,161)( 82,164)( 83,163)
( 84,162)( 85,157)( 86,160)( 87,159)( 88,158)( 89,153)( 90,156)( 91,155)
( 92,154)( 93,149)( 94,152)( 95,151)( 96,150)( 97,145)( 98,148)( 99,147)
(100,146)(101,141)(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)
(108,138)(109,133)(110,136)(111,135)(112,134)(113,129)(114,132)(115,131)
(116,130)(117,125)(118,128)(119,127)(120,126)(122,124);
poly := sub<Sym(236)|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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope