ΓL(1,𝔽₈)×#462 #13054993

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

Aliases

direct product	ΓL(1,𝔽₈)×#462
unambiguous semidirect product	(C₇×#462)⋊C₃

Splittings

Fitting, Hall	C₇×#462 ⋊ C₃
Sylow	C₇ ⋊ C₃×#462

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#462	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×D₄
hypercenter	#462	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#462
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

216 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

(2,3,4)(5,7,6)(9,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21);(1,2,3,5,4,6,7)(8,9,10,11);(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4282);