ΓL(1,𝔽₈)×#576 #13060737

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

ΓL(1,𝔽₈)×#576

Splittings

Fitting, Hall	C₇×#576 ⋊ C₃
Sylow	C₇ ⋊ C₃×#576

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#576	C₃
socle	C₂×C₇	C₂²×C₃×D₄
hypercenter	#576	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#576
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

536 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 237 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 — 4 generators on 23 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#3585, ΓL(1,𝔽₈)×#3292, AGL(1,𝔽₇)×#576, …
maximal normal subgroup of	ΓL(1,𝔽₈)×#3585, ΓL(1,𝔽₈)×#3292, AGL(1,𝔽₇)×#576, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10026);