ΓL(1,𝔽₈)×#544 #13060705

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

Aliases

direct product	ΓL(1,𝔽₈)×#544
unambiguous semidirect product	(C₇×#544)⋊C₃

Splittings

Fitting, Hall	C₇×#544 ⋊ C₃
Sylow	C₇ ⋊ C₃×#544

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#544	C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	#544	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#544
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #544

412 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 243 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)(11,13)(12,15)(16,17,19,21)(18,22,23,20);(1,2,3,5,4,6,7)(8,9,11,13)(10,12,15,14);(8,9)(10,12)(11,13)(14,15)(16,18,19,23)(17,20,21,22);(8,10)(9,12)(11,15)(13,14)

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, 9994);