ΓL(1,𝔽₈)×#433 #13054964

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

Aliases

direct product	ΓL(1,𝔽₈)×#433
unambiguous semidirect product	(C₇×#433)⋊C₃

Splittings

Fitting, Hall	C₇×#433 ⋊ C₃
Sylow	C₇ ⋊ C₃×#433

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
commutator, socle	C₂²×C₇	C₂²×C₃×C₄
Fitting	C₇×#433	C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#433	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#433
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

192 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

(2,3,4)(5,7,6)(8,9)(10,13)(11,12)(14,15)(16,17,19,22,20,23,26,29)(18,21,25,28,24,27,30,31);(1,2,3,5,4,6,7)(9,12)(13,15)(16,18)(17,21)(19,25)(20,24)(22,28)(23,27)(26,30)(29,31);(8,10,11,14)(9,13,12,15)(18,24)(21,27)(25,30)(28,31)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4253);