D₂₃×ΓL(1,𝔽₈)⋊D₇ #27214151

order	13524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₂₃×ΓL(1,𝔽₈)⋊D₇
unambiguous semidirect product	(ΓL(1,𝔽₈)×D₂₃)⋊D₇

Splittings

Fitting, socle, derived, Hall	C₇²×C₂₃ ⋊ C₂²×C₃
Hall	D₂₃×C₇⋊D₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈)×C₂₃ ⋊ C₂²
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×D₂₃
Sylow	C₇² ⋊ C₂×C₃×D₂₃
Sylow	C₂₃ ⋊ C₂×ΓL(1,𝔽₈)⋊D₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₂₃	C₂²×C₃
LCS	G ▷ C₇²×C₂₃
UCS	C₁

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 483

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(25,26)(27,28)(29,30)(32,35,36,33,34,37);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25,27,29,30,28,26)(31,32,34,37,36,35,33)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13524, 85);

D23×ΓL(1,𝔽8)⋊D7 #27214151