C₂²×Q₁₆×ΓL(1,𝔽₈) #13060732

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

Fitting, Hall	C₂²×C₇×Q₁₆ ⋊ C₃
Sylow	C₇ ⋊ C₂²×C₃×Q₁₆

516 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 267 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 — 4 generators on 27 points:

(1,2)(22,23,24)(25,27,26);(3,4)(21,22,23,25,24,26,27);(5,6,7,8,9,10,11,12)(13,20,19,18,17,16,15,14);(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, 10021);