Q₈×Dic₇ #6508

order	224
automorphism	Aut(G) = C₂²×AGL(1,𝔽₇)×#1020 (order 16128) · Inn(G) = C₂²×D₇ · Out(G) = C₂²×C₃×S₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₇ ⋊ C₄×Q₈

70 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 140);