ΓL(1,𝔽₈)×#473 #13055004

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

Aliases

direct product	ΓL(1,𝔽₈)×#473
unambiguous semidirect product	(C₇×#473)⋊C₃

Splittings

Fitting, Hall	C₇×#473 ⋊ C₃
Sylow	C₇ ⋊ C₃×#473

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#121
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#473	C₃
socle	C₂×C₇	C₃×#121
prime radical	#41×ΓL(1,𝔽₈)	C₂²
hypercenter	#473	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#473
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #473

200 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(2,3,4)(5,7,6)(8,9,14,19)(10,20,24,30)(11,25,27,33)(12,18,13,17)(15,23,31,22)(16,28,32,26)(21,35,37,29)(34,38,39,36);(1,2,3,5,4,6,7)(8,10,13,23,14,24,12,22)(9,15,18,30,19,31,17,20)(11,21,28,38,27,37,26,36)(16,29,33,39,32,35,25,34);(8,11)(9,16)(10,21)(12,26)(13,28)(14,27)(15,29)(17,25)(18,33)(19,32)(20,34)(22,36)(23,38)(24,37)(30,39)(31,35)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4293);