ΓL(1,𝔽₈)×#454 #13054985

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

Aliases

direct product	ΓL(1,𝔽₈)×#454
unambiguous semidirect product	(C₇×#454)⋊C₃

Splittings

Fitting, Hall	C₇×#454 ⋊ C₃
Sylow	C₇ ⋊ C₃×#454

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#120
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#454	C₃
socle	C₂×C₇	C₃×#120
hypercenter	#454	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#454
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #454

292 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 81 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)(11,18)(14,22)(15,19)(16,20)(21,23);(1,2,3,5,4,6,7)(8,9)(10,14)(11,19)(12,15)(13,18)(16,21)(17,23)(20,22);(8,10,12,17)(9,14,15,23)(11,20,18,16)(13,21,19,22)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#3288, ΓL(1,𝔽₈)×#3018, ΓL(1,𝔽₈)×#3021, …
maximal normal subgroup of	ΓL(1,𝔽₈)×#3288, ΓL(1,𝔽₈)×#3018, ΓL(1,𝔽₈)×#3021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4274);