C₄×Q₁₆×ΓL(1,𝔽₈) #13054970

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

Aliases

direct product	C₄×Q₁₆×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₄×C₇×Q₁₆)⋊C₃

Fitting, Hall	C₄×C₇×Q₁₆ ⋊ C₃
Sylow	C₇ ⋊ C₃×C₄×Q₁₆

220 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 448

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4259);