D₁₁₄₇₃ #37525770

order	22946
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₄₇₃
unambiguous semidirect product	C₇⋊D₁₆₃₉
unambiguous semidirect product	C₁₁⋊D₁₀₄₃
unambiguous semidirect product	C₁₄₉⋊D₇₇

Splittings

Fitting, socle, derived, Hall	C₇×C₁₁×C₁₄₉ ⋊ C₂
Hall	C₁₁×C₁₄₉ ⋊ D₇
Hall	C₇×C₁₄₉ ⋊ D₁₁
Sylow	C₁₄₉ ⋊ D₇₇
Hall	C₇×C₁₁ ⋊ D₁₄₉
Sylow	C₁₁ ⋊ D₁₀₄₃
Sylow	C₇ ⋊ D₁₆₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₁×C₁₄₉	C₂
LCS	G ▷ C₇×C₁₁×C₁₄₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 167
transitive degree	μ_t(G) = 11473

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

Polycyclic presentation:

2,7,11,149|b^a=b^6,c^a=c^10,d^a=d^148

Conjugacy classes & character table

5738 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22946, 8);

D11473 #37525770