D₁₁₆₂₇ #37551374

order	23254
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₆₂₇
unambiguous semidirect product	C₇⋊D₁₆₆₁
unambiguous semidirect product	C₁₁⋊D₁₀₅₇
unambiguous semidirect product	C₁₅₁⋊D₇₇

Splittings

Fitting, socle, derived, Hall	C₇×C₁₁×C₁₅₁ ⋊ C₂
Hall	C₁₁×C₁₅₁ ⋊ D₇
Hall	C₇×C₁₅₁ ⋊ D₁₁
Sylow	C₁₅₁ ⋊ D₇₇
Hall	C₇×C₁₁ ⋊ D₁₅₁
Sylow	C₁₁ ⋊ D₁₀₅₇
Sylow	C₇ ⋊ D₁₆₆₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₁×C₁₅₁	C₂
LCS	G ▷ C₇×C₁₁×C₁₅₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 169
transitive degree	μ_t(G) = 11627

A minimal faithful permutation representation — 2 generators on 169 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);(2,7)(3,6)(4,5)(9,18)(10,17)(11,16)(12,15)(13,14)(20,169)(21,168)(22,167)(23,166)(24,165)(25,164)(26,163)(27,162)(28,161)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)

Polycyclic presentation:

2,7,11,151|b^a=b^6,c^a=c^10,d^a=d^150

Conjugacy classes & character table

5815 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23254, 8);

D11627 #37551374