ΓL(1,𝔽₈)×#399 #13054930

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

Aliases

direct product

ΓL(1,𝔽₈)×#399

Splittings

Fitting, Hall	C₇×#399 ⋊ C₃
Sylow	C₇ ⋊ C₃×#399

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂³×C₇	C₂³×C₃
Fitting	C₇×#399	C₃
prime radical	C₂⁴×ΓL(1,𝔽₈)	C₂²
hypercenter	#399	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#399
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #399

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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 — 3 generators on 23 points:

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

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, 4219);