ΓL(1,𝔽₈)×#471 #13055002

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

Aliases

direct product	ΓL(1,𝔽₈)×#471
unambiguous semidirect product	(C₇×#471)⋊C₃

Splittings

Fitting, Hall	C₇×#471 ⋊ C₃
Sylow	C₇ ⋊ C₃×#471

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#121
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#471	C₃
socle	C₂×C₇	C₃×#121
prime radical	ΓL(1,𝔽₈)×AGL(1,ℤ/8)	C₂
hypercenter	#471	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#471
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #471

208 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 112

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#3291, ΓL(1,𝔽₈)×#3288
maximal normal subgroup of	ΓL(1,𝔽₈)×#3291, ΓL(1,𝔽₈)×#3288

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4291);