ΓL(1,𝔽₈)×#470 #13055001

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

Aliases

direct product	ΓL(1,𝔽₈)×#470
unambiguous semidirect product	(C₇×#470)⋊C₃

Splittings

Fitting, Hall	C₇×#470 ⋊ C₃
Sylow	C₇ ⋊ C₃×#470

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×#470	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂³×ΓL(1,𝔽₈)	D₄
hypercenter	#470	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#470
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

220 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

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, 4290);