ΓL(1,𝔽₈)×#413 #13054944

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

Aliases

direct product	ΓL(1,𝔽₈)×#413
unambiguous semidirect product	(C₇×#413)⋊C₃

Splittings

Fitting, Hall	C₇×#413 ⋊ C₃
Sylow	C₇ ⋊ C₃×#413

Subgroups

	subgroup	G/H
center	C₄	C₂²⋊C₄×ΓL(1,𝔽₈)
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#413	C₃
socle	C₂×C₇	C₂×C₃×C₂²⋊C₄
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#413	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#413
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

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

normal subgroups only

This group has 300 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)(8,9,12,17,13,16,11,15)(10,18,19,23,20,14,21,22);(1,2,3,5,4,6,7)(8,10)(9,14)(11,19)(12,21)(13,20)(15,22)(16,18)(17,23);(8,11,13,12)(9,15,16,17)(10,19,20,21)(14,22,18,23)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#1914, ΓL(1,𝔽₈)×#3018, ΓL(1,𝔽₈)×#3021, …
maximal normal subgroup of	ΓL(1,𝔽₈)×#1914, ΓL(1,𝔽₈)×#3018, ΓL(1,𝔽₈)×#3021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4233);