D₂₆₁₉₅ #72289504

order	52390
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₁₉₅

Splittings

Fitting, derived, Hall	C₅×C₃₁×C₁₆₉ ⋊ C₂
Hall	C₃₁×C₁₆₉ ⋊ D₅
Hall	C₅×C₁₆₉ ⋊ D₃₁
Sylow	C₁₆₉ ⋊ D₁₅₅
Hall	C₅×C₃₁ ⋊ D₁₆₉
Sylow	C₃₁ ⋊ D₈₄₅
Sylow	C₅ ⋊ D₅₂₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₃₁×C₁₆₉	C₂
Frattini	C₁₃	D₂₀₁₅
socle	C₅×C₁₃×C₃₁	D₁₃
LCS	G ▷ C₅×C₃₁×C₁₆₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 205
transitive degree	μ_t(G) = 26195

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

Polycyclic presentation:

2,5,13,31,13|c^13=e,b^a=b^4,c^a=c^12e^12,d^a=d^30,e^a=e^12

Conjugacy classes & character table

13099 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, 5, 13, 31, 13]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4);;
gap> SetConjugate(coll, 3, 1, F.3^12*F.5^12);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26195 #72289504