C₄×C₂²⋊C₄×ΓL(1,𝔽₈) #13054908

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product	C₄×C₂²⋊C₄×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₄×C₇×C₂²⋊C₄)⋊C₃

Fitting, Hall	C₄×C₇×C₂²⋊C₄ ⋊ C₃
Sylow	C₇ ⋊ C₃×C₄×C₂²⋊C₄

516 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 219 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)(14,15,16)(17,19,18);(5,6,8,10)(7,9,11,12)(13,14,15,17,16,18,19);(6,9)(10,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4197);