ΓL(1,𝔽₈)×#350 #13051078

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	ΓL(1,𝔽₈)×#350
unambiguous semidirect product	(C₇×#350)⋊C₃

Splittings

Fitting, Hall	C₇×#350 ⋊ C₃
Sylow	C₇ ⋊ C₃×#350

Subgroups

	subgroup	G/H
center	C₈	D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂×C₃×C₈
Frattini	C₂×C₈	C₂²×ΓL(1,𝔽₈)
Fitting	C₇×#350	C₃
socle	C₂×C₇	C₃×C₂²⋊C₈
prime radical	#41×ΓL(1,𝔽₈)	C₄
cosocle	C₂×C₇×C₈	C₂²×C₃
hypercenter	#350	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#350
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

112 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(2,3,4)(5,7,6)(8,9,12,17,13,18,27,35,14,19,28,36,11,16,25,33)(10,20,22,37,21,31,38,34,23,32,26,39,24,15,29,30);(1,2,3,5,4,6,7)(8,10)(9,15)(11,24)(12,26)(13,21)(14,23)(16,32)(17,34)(18,20)(19,31)(22,28)(25,38)(27,29)(30,36)(33,37)(35,39)

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, 367);