ΓL(1,𝔽₈)×#555 #13060716

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product	ΓL(1,𝔽₈)×#555
unambiguous semidirect product	(C₇×#555)⋊C₃

Splittings

Fitting, Hall	C₇×#555 ⋊ C₃
Sylow	C₇ ⋊ C₃×#555

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#555	C₃
hypercenter	#555	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#555
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #555

472 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 225 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)(11,15)(12,13)(17,18)(21,22);(1,2,3,5,4,6,7)(9,13)(11,14)(16,17,20,18)(19,22,23,21);(8,10,12,15)(9,14,13,11)(16,18,20,17)(19,22,23,21);(8,11,12,14)(9,15,13,10)(16,19,20,23)(17,21,18,22)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10005);