ΓL(1,𝔽₈)×#3291 #72229607

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

ΓL(1,𝔽₈)×#3291

Splittings

Fitting, Hall	C₇×#3291 ⋊ C₃
Sylow	C₇ ⋊ C₃×#3291

Subgroups

	subgroup	G/H
center	C₂	D₄²×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#3291	C₃
socle	C₂×C₇	C₃×D₄²
hypercenter	#3291	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#3291
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3291

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 23 points:

(2,3,4)(5,7,6)(8,9,12,15)(10,17,19,22)(11,13,20,16)(14,18,21,23);(1,2,4,6,3,5,7)(9,13)(10,18)(11,20)(15,16)(17,22)(19,23);(9,13,15,16)(14,17,21,22);(8,10,11,18,12,19,20,23)(9,14,16,22,15,21,13,17)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4)(5,7,6)(8,9,12,15)(10,17,19,22)(11,13,20,16)(14,18,21,23), (1,2,4,6,3,5,7)(9,13)(10,18)(11,20)(15,16)(17,22)(19,23), (9,13,15,16)(14,17,21,22), (8,10,11,18,12,19,20,23)(9,14,16,22,15,21,13,17));

ΓL(1,𝔽8)×#3291 #72229607