ΓL(1,𝔽₈)×#563 #13060724

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

Aliases

direct product	ΓL(1,𝔽₈)×#563
unambiguous semidirect product	(C₇×#563)⋊C₃

Splittings

Fitting, Hall	C₇×#563 ⋊ C₃
Sylow	C₇ ⋊ C₃×#563

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#563	C₃
prime radical	C₂×D₄×ΓL(1,𝔽₈)	C₂²
hypercenter	#563	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#563
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #563

392 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 213 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10013);