D₁₁₅₆₅ #37528904

order	23130
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₅₆₅
unambiguous semidirect product	C₅⋊D₂₃₁₃
unambiguous semidirect product	C₉⋊D₁₂₈₅
unambiguous semidirect product	C₂₅₇⋊D₄₅

Splittings

Fitting, derived, Hall	C₅×C₉×C₂₅₇ ⋊ C₂
Hall	C₉×C₂₅₇ ⋊ D₅
Hall	C₅×C₂₅₇ ⋊ D₉
Sylow	C₂₅₇ ⋊ D₄₅
Hall	C₅×C₉ ⋊ D₂₅₇
Sylow	C₉ ⋊ D₁₂₈₅
Sylow	C₅ ⋊ D₂₃₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₉×C₂₅₇	C₂
Frattini	C₃	D₃₈₅₅
socle	C₃×C₅×C₂₅₇	D₃
LCS	G ▷ C₅×C₉×C₂₅₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 271
transitive degree	μ_t(G) = 11565

A minimal faithful permutation representation — 2 generators on 271 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,271)(17,270)(18,269)(19,268)(20,267)(21,266)(22,265)(23,264)(24,263)(25,262)(26,261)(27,260)(28,259)(29,258)(30,257)(31,256)(32,255)(33,254)(34,253)(35,252)(36,251)(37,250)(38,249)(39,248)(40,247)(41,246)(42,245)(43,244)(44,243)(45,242)(46,241)(47,240)(48,239)(49,238)(50,237)(51,236)(52,235)(53,234)(54,233)(55,232)(56,231)(57,230)(58,229)(59,228)(60,227)(61,226)(62,225)(63,224)(64,223)(65,222)(66,221)(67,220)(68,219)(69,218)(70,217)(71,216)(72,215)(73,214)(74,213)(75,212)(76,211)(77,210)(78,209)(79,208)(80,207)(81,206)(82,205)(83,204)(84,203)(85,202)(86,201)(87,200)(88,199)(89,198)(90,197)(91,196)(92,195)(93,194)(94,193)(95,192)(96,191)(97,190)(98,189)(99,188)(100,187)(101,186)(102,185)(103,184)(104,183)(105,182)(106,181)(107,180)(108,179)(109,178)(110,177)(111,176)(112,175)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,168)(120,167)(121,166)(122,165)(123,164)(124,163)(125,162)(126,161)(127,160)(128,159)(129,158)(130,157)(131,156)(132,155)(133,154)(134,153)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)

Polycyclic presentation:

2,3,5,257,3|b^3=e^2,b^a=b^2e,c^a=c^4,d^a=d^256,e^a=e^2

Conjugacy classes & character table

5784 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23130, 8);

D11565 #37528904