ΓL(1,𝔽₈)×#395 #13054926

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

Aliases

direct product

ΓL(1,𝔽₈)×#395

Splittings

Fitting, Hall	C₇×#395 ⋊ C₃
Sylow	C₇ ⋊ C₃×#395

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂³×C₇	C₂³×C₃
Fitting	C₇×#395	C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	#395	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#395
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #395

364 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

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

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, 4215);