ΓL(1,𝔽₈)×#430 #13054961

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

Aliases

direct product	ΓL(1,𝔽₈)×#430
unambiguous semidirect product	(C₇×#430)⋊C₃

Splittings

Fitting, Hall	C₇×#430 ⋊ C₃
Sylow	C₇ ⋊ C₃×#430

Subgroups

	subgroup	G/H
center	C₄	C₂×D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#430	C₃
socle	C₂×C₇	C₂×C₃×C₄⋊C₄
prime radical	C₂³×ΓL(1,𝔽₈)	Q₈
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#430	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#430
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

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

normal subgroups only

This group has 204 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) = 112

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#2956, AGL(1,𝔽₇)×#430
maximal normal subgroup of	ΓL(1,𝔽₈)×#2956, AGL(1,𝔽₇)×#430

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4250);