ΓL(1,𝔽₈)×#472 #13055003

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	ΓL(1,𝔽₈)×#472
unambiguous semidirect product	(C₇×#472)⋊C₃

Splittings

Fitting, Hall	C₇×#472 ⋊ C₃
Sylow	C₇ ⋊ C₃×#472

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#121
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#472	C₃
socle	C₂×C₇	C₃×#121
hypercenter	#472	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#472
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #472

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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)(9,13)(10,17)(11,14)(16,21)(18,19)(20,22);(1,2,3,5,4,6,7)(8,9,14,21,12,16,11,13)(10,15,17,22,19,23,18,20);(8,10)(9,15)(11,18)(12,19)(13,20)(14,17)(16,23)(21,22)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#3292, ΓL(1,𝔽₈)×#3288
maximal normal subgroup of	ΓL(1,𝔽₈)×#3292, ΓL(1,𝔽₈)×#3288

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4292);