ΓL(1,𝔽₈)×#540 #13060701

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

Aliases

direct product	ΓL(1,𝔽₈)×#540
unambiguous semidirect product	(C₇×#540)⋊C₃

Splittings

Fitting, Hall	C₇×#540 ⋊ C₃
Sylow	C₇ ⋊ C₃×#540

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#540	C₃
hypercenter	#540	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#540
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #540

480 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 219 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)(8,9)(10,14)(11,12)(13,15)(17,20)(18,22);(1,2,3,5,4,6,7)(9,12)(13,14)(16,17,19,20)(18,21,22,23);(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,20)(22,23);(8,10,11,15)(9,13,12,14)(16,18,19,22)(17,21,20,23)

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, 9990);