ΓL(1,𝔽₈)×#408 #13054939

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	ΓL(1,𝔽₈)×#408
unambiguous semidirect product	(C₇×#408)⋊C₃

Splittings

Fitting, Hall	C₇×#408 ⋊ C₃
Sylow	C₇ ⋊ C₃×#408

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
commutator, socle	C₂²×C₇	C₂²×C₃×C₄
Fitting	C₇×#408	C₃
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#408	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#408
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

316 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 224

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4228);