ΓL(1,𝔽₈)×#481 #13055012

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

Aliases

direct product	ΓL(1,𝔽₈)×#481
unambiguous semidirect product	(C₇×#481)⋊C₃

Splittings

Fitting, Hall	C₇×#481 ⋊ C₃
Sylow	C₇ ⋊ C₃×#481

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#481	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	ΓL(1,𝔽₈)×#121	C₂
hypercenter	#481	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#481
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #481

228 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 81 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)(8,9)(10,13)(11,12)(14,15)(16,17)(18,20)(19,21)(22,23);(1,2,3,5,4,6,7)(9,12)(13,15)(17,20)(21,22);(8,10,11,14)(9,13,12,15)(17,21)(19,23)(20,22)

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4301);