ΓL(1,𝔽₈)×#400 #13054931

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	ΓL(1,𝔽₈)×#400
unambiguous semidirect product	(C₇×#400)⋊C₃

Splittings

Fitting, Hall	C₇×#400 ⋊ C₃
Sylow	C₇ ⋊ C₃×#400

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂³×C₇	C₂³×C₃
Fitting	C₇×#400	C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	#400	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#400
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #400

268 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 27 points:

(2,3,4)(5,7,6)(14,18)(16,19)(20,21,23,25)(22,26,27,24);(1,2,3,5,4,6,7)(9,11)(12,13,15,17)(14,19,18,16)(20,21)(22,26)(23,25)(24,27);(8,9)(10,11)(12,14)(13,16)(15,18)(17,19)(20,22,23,27)(21,24,25,26)

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, 4220);