ΓL(1,𝔽₈)×D₄⋊C₈ #13051053

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

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

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

164 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

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

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 342);