ΓL(1,𝔽₈)×#2230 #72229662

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

ΓL(1,𝔽₈)×#2230

Splittings

Fitting, Hall	C₇×#2230 ⋊ C₃
Sylow	C₇ ⋊ C₃×#2230

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#476
commutator, cosocle	C₂×C₇×C₈	C₂³×C₃
Frattini	C₂×C₈	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#2230	C₃
socle	C₂×C₇	C₃×#476
prime radical	ΓL(1,𝔽₈)×#135	C₂²
hypercenter	#2230	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#2230
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2230

252 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

115 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)(8,9,12,16)(10,13,20,23)(11,21,18,14)(15,17,22,19), (1,2,4,6,3,5,7)(9,13,16,23)(10,17)(11,18)(14,15,21,22)(19,20), (9,14,16,21)(13,22,23,15));