C₂×C₄×C₈×ΓL(1,𝔽₈) #13054933

order	1344
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product	C₂×C₄×C₈×ΓL(1,𝔽₈)
unambiguous semidirect product	(C₂×C₄×C₇×C₈)⋊C₃

Fitting, Hall	C₂×C₄×C₇×C₈ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₄×C₈

324 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4222);