ΓL(1,𝔽₈)×#410 #13054941

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

Aliases

direct product	ΓL(1,𝔽₈)×#410
unambiguous semidirect product	(C₇×#410)⋊C₃

Splittings

Fitting, Hall	C₇×#410 ⋊ C₃
Sylow	C₇ ⋊ C₃×#410

Subgroups

	subgroup	G/H
center	C₄	C₂²⋊C₄×ΓL(1,𝔽₈)
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#410	C₃
socle	C₂×C₇	C₂×C₃×C₂²⋊C₄
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	#410	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#410
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

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

normal subgroups only

This group has 316 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)(10,16,18,13)(11,14)(12,15)(17,22,23,20)(19,21);(1,2,3,5,4,6,7)(8,10)(9,13)(11,17)(12,18)(14,20)(15,16)(19,23)(21,22);(8,11,12,19)(9,14,15,21)(10,17,18,23)(13,20,16,22)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#2008, ΓL(1,𝔽₈)×#1906, AGL(1,𝔽₇)×#410, …
maximal normal subgroup of	ΓL(1,𝔽₈)×#2008, ΓL(1,𝔽₈)×#1906, AGL(1,𝔽₇)×#410, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4230);