ΓL(1,𝔽₈)×#405 #13054936

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product	ΓL(1,𝔽₈)×#405
unambiguous semidirect product	(C₇×#405)⋊C₃

Splittings

Fitting, Hall	C₇×#405 ⋊ C₃
Sylow	C₇ ⋊ C₃×#405

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#405	C₃
socle	C₂²×C₇	C₂²×C₃×C₄
prime radical	C₂³×ΓL(1,𝔽₈)	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#405	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

260 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 171 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)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27);(1,2,3,5,4,6,7)(8,9,10,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(14,20)(17,23)(21,26)(24,27)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4225);