C₂×ΓL(1,𝔽₈)×#108 #13054960

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

Aliases

direct product	C₂×ΓL(1,𝔽₈)×#108
unambiguous semidirect product	(C₂×C₇×#108)⋊C₃

Fitting, Hall	C₂×C₇×#108 ⋊ C₃
Sylow	C₇ ⋊ C₂×C₃×#108

212 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

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

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4249);