ΓL(1,𝔽₈)×#386 #13054917

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product	ΓL(1,𝔽₈)×#386
unambiguous semidirect product	(C₇×#386)⋊C₃

Fitting, Hall	C₇×#386 ⋊ C₃
Sylow	C₇ ⋊ C₃×#386

572 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4206);