ΓL(1,𝔽₈)×#541 #13060702

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

Aliases

direct product	ΓL(1,𝔽₈)×#541
unambiguous semidirect product	(C₇×#541)⋊C₃

Splittings

Fitting, Hall	C₇×#541 ⋊ C₃
Sylow	C₇ ⋊ C₃×#541

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#541	C₃
prime radical	C₂³×ΓL(1,𝔽₈)	C₂³
hypercenter	#541	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#541
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #541

384 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 219 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)(10,14)(12,15)(16,17,20,23)(18,21,26,24)(19,27,28,22)(25,30,31,29);(1,2,3,5,4,6,7)(8,9,11,13)(10,12,14,15)(16,17)(18,24)(19,22)(20,23)(21,26)(25,30)(27,28)(29,31);(8,10)(9,12)(11,14)(13,15)(16,18)(17,21)(19,25)(20,26)(22,29)(23,24)(27,30)(28,31);(16,19,20,28)(17,22,23,27)(18,25,26,31)(21,29,24,30)

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, 9991);