D₂₉₉₁₃ #72294852

order	59826
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₉₉₁₃

Splittings

Fitting, derived, Hall	C₃×C₅₉×C₁₆₉ ⋊ C₂
Hall	C₅₉×C₁₆₉ ⋊ D₃
Hall	C₃×C₁₆₉ ⋊ D₅₉
Hall	C₃×C₅₉ ⋊ D₁₆₉
Sylow	C₁₆₉ ⋊ D₁₇₇
Sylow	C₅₉ ⋊ D₅₀₇
Sylow	C₃ ⋊ D₉₉₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₅₉×C₁₆₉	C₂
Frattini	C₁₃	D₂₃₀₁
socle	C₃×C₁₃×C₅₉	D₁₃
LCS	G ▷ C₃×C₅₉×C₁₆₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 231
transitive degree	μ_t(G) = 29913

A minimal faithful permutation representation — 2 generators on 231 points:

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

Polycyclic presentation:

2,3,13,59,13|c^13=e,b^a=b^2,c^a=c^12e^12,d^a=d^58,e^a=e^12

Conjugacy classes & character table

14958 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 3, 13, 59, 13]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^12*F.5^12);;
gap> SetConjugate(coll, 4, 1, F.4^58);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29913 #72294852