ΓL(1,𝔽₈)×#441 #13054972

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

Aliases

direct product	ΓL(1,𝔽₈)×#441
unambiguous semidirect product	(C₇×#441)⋊C₃

Splittings

Fitting, Hall	C₇×#441 ⋊ C₃
Sylow	C₇ ⋊ C₃×#441

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×#441	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#441	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

216 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 105 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)(8,9,11,10)(12,13)(14,20)(15,23)(16,18)(17,21)(19,25)(22,27)(24,26);(1,2,3,5,4,6,7)(12,14,16,22)(13,17,18,26)(15,24,25,21)(19,27,23,20);(8,10,11,9)(13,18)(17,26)(19,23)(20,27)

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, 4261);