ΓL(1,𝔽₈)×#451 #13054982

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

Aliases

direct product	ΓL(1,𝔽₈)×#451
unambiguous semidirect product	(C₇×#451)⋊C₃

Splittings

Fitting, Hall	C₇×#451 ⋊ C₃
Sylow	C₇ ⋊ C₃×#451

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×#451	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂³×ΓL(1,𝔽₈)	D₄
hypercenter	#451	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#451
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 224

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

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

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, 4271);