ΓL(1,𝔽₈)×#464 #13054995

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

Aliases

direct product	ΓL(1,𝔽₈)×#464
unambiguous semidirect product	(C₇×#464)⋊C₃

Splittings

Fitting, Hall	C₇×#464 ⋊ C₃
Sylow	C₇ ⋊ C₃×#464

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×#464	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₂²
hypercenter	#464	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#464
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

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

normal subgroups only

This group has 224 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) = 27
transitive degree	μ_t(G) = 224

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

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

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, 4284);