ΓL(1,𝔽₈)×#578 #13060739

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

ΓL(1,𝔽₈)×#578

Splittings

Fitting, Hall	C₇×#578 ⋊ C₃
Sylow	C₇ ⋊ C₃×#578

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#578	C₃
socle	C₂×C₇	C₂²×C₃×D₄
hypercenter	#578	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#578
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

496 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 237 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 — 4 generators on 39 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10028);