ΓL(1,𝔽₈)×#421 #13054952

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	ΓL(1,𝔽₈)×#421
unambiguous semidirect product	(C₇×#421)⋊C₃

Splittings

Fitting, Hall	C₇×#421 ⋊ C₃
Sylow	C₇ ⋊ C₃×#421

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×#421	C₃
socle	C₂×C₇	C₂×C₃×C₂²⋊C₄
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#421	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#421
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

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

normal subgroups only

This group has 308 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)(8,9,12,15,13,16,11,20)(10,17,19,14,18,23,21,22);(1,2,3,5,4,6,7)(8,10)(9,14)(11,21)(12,19)(13,18)(15,23)(16,22)(17,20);(10,18)(14,22)(17,23)(19,21)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#3019, ΓL(1,𝔽₈)×#2008, ΓL(1,𝔽₈)×#1906, …
maximal normal subgroup of	ΓL(1,𝔽₈)×#3019, ΓL(1,𝔽₈)×#2008, ΓL(1,𝔽₈)×#1906, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4241);