ΓL(1,𝔽₈)×#443 #13054974

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

Aliases

direct product	ΓL(1,𝔽₈)×#443
unambiguous semidirect product	(C₇×#443)⋊C₃

Splittings

Fitting, Hall	C₇×#443 ⋊ C₃
Sylow	C₇ ⋊ C₃×#443

Subgroups

	subgroup	G/H
center	C₈	D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#443	C₃
socle	C₂×C₇	C₃×C₄×D₄
prime radical	ΓL(1,𝔽₈)×#135	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#443	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#443
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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)(8,9,12,16,13,10,17,15)(11,14,19,22,20,18,23,21);(1,2,3,5,4,6,7)(8,10)(9,13)(11,18)(12,15)(14,20)(16,17)(19,21)(22,23);(8,11,12,19,13,20,17,23)(9,14,16,22,10,18,15,21)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#443, ΓL(1,𝔽₈)×#3292, #72229599, …
maximal normal subgroup of	AGL(1,𝔽₇)×#443, ΓL(1,𝔽₈)×#3292, #72229599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4263);