ΓL(1,𝔽₈)×#446 #13054977

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	ΓL(1,𝔽₈)×#446
unambiguous semidirect product	(C₇×#446)⋊C₃

Splittings

Fitting, Hall	C₇×#446 ⋊ C₃
Sylow	C₇ ⋊ C₃×#446

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
commutator, socle	C₂²×C₇	C₂²×C₃×C₄
Fitting	C₇×#446	C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#446	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#446
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

188 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 448

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4266);