D₁₁₇₆₅ #37555188

order	23530
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇₆₅
unambiguous semidirect product	C₅⋊D₂₃₅₃
unambiguous semidirect product	C₁₃⋊D₉₀₅
unambiguous semidirect product	C₁₈₁⋊D₆₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₃×C₁₈₁ ⋊ C₂
Hall	C₁₃×C₁₈₁ ⋊ D₅
Hall	C₅×C₁₈₁ ⋊ D₁₃
Sylow	C₁₈₁ ⋊ D₆₅
Hall	C₅×C₁₃ ⋊ D₁₈₁
Sylow	C₁₃ ⋊ D₉₀₅
Sylow	C₅ ⋊ D₂₃₅₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₃×C₁₈₁	C₂
LCS	G ▷ C₅×C₁₃×C₁₈₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 199
transitive degree	μ_t(G) = 11765

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

Polycyclic presentation:

2,5,13,181|b^a=b^4,c^a=c^12,d^a=d^180

Conjugacy classes & character table

5884 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23530, 9);

D11765 #37555188