D₃₂₆₆₁ #72304144

order	65322
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₂₆₆₁

Splittings

Fitting, derived, Hall	C₉×C₁₉×C₁₉₁ ⋊ C₂
Hall	C₁₉×C₁₉₁ ⋊ D₉
Hall	C₉×C₁₉₁ ⋊ D₁₉
Sylow	C₁₉₁ ⋊ D₁₇₁
Hall	C₉×C₁₉ ⋊ D₁₉₁
Sylow	C₁₉ ⋊ D₁₇₁₉
Sylow	C₉ ⋊ D₃₆₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₉×C₁₉₁	C₂
Frattini	C₃	D₁₀₈₈₇
socle	C₃×C₁₉×C₁₉₁	D₃
LCS	G ▷ C₉×C₁₉×C₁₉₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 219
transitive degree	μ_t(G) = 32661

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

Polycyclic presentation:

2,3,19,191,3|b^3=e,b^a=b^2e^2,c^a=c^18,d^a=d^190,e^a=e^2

Conjugacy classes & character table

16332 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, 19, 191, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^18);;
gap> SetConjugate(coll, 4, 1, F.4^190);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32661 #72304144