ΓL(1,𝔽₈)×#485 #13055016

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	ΓL(1,𝔽₈)×#485
unambiguous semidirect product	(C₇×#485)⋊C₃

Splittings

Fitting, Hall	C₇×#485 ⋊ C₃
Sylow	C₇ ⋊ C₃×#485

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#485	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂³×ΓL(1,𝔽₈)	D₄
hypercenter	#485	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#485
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

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

normal subgroups only

This group has 192 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 4305);