C₈×D₄×ΓL(1,𝔽₈) #13054965

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₈×D₄×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₇×C₈×D₄)⋊C₃

Fitting, Hall	C₇×C₈×D₄ ⋊ C₃
Sylow	C₇ ⋊ C₃×C₈×D₄

268 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 135 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4254);