ΓL(1,𝔽₈)×#398 #13054929

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

Aliases

direct product	ΓL(1,𝔽₈)×#398
unambiguous semidirect product	(C₇×#398)⋊C₃

Fitting, Hall	C₇×#398 ⋊ C₃
Sylow	C₇ ⋊ C₃×#398

300 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4218);