ΓL(1,𝔽₈)×#562 #13060723

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	ΓL(1,𝔽₈)×#562
unambiguous semidirect product	(C₇×#562)⋊C₃

Splittings

Fitting, Hall	C₇×#562 ⋊ C₃
Sylow	C₇ ⋊ C₃×#562

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#562	C₃
hypercenter	#562	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#562
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #562

440 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 213 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) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 23 points:

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

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, 10012);