ΓL(1,𝔽₈)×#419 #13054950

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	ΓL(1,𝔽₈)×#419
unambiguous semidirect product	(C₇×#419)⋊C₃

Splittings

Fitting, Hall	C₇×#419 ⋊ C₃
Sylow	C₇ ⋊ C₃×#419

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#419	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂³×ΓL(1,𝔽₈)	D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#419	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#419
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

300 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 123 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)(14,21)(15,19)(16,25)(17,20)(18,23)(22,26)(24,27);(1,2,3,5,4,6,7)(12,14,17,24)(13,18,20,26)(15,23,16,22)(19,27,25,21);(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,21,26,27)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4239);