order	224
automorphism	Aut(G) = C₂×AGL(1,ℤ/8)×AGL(1,𝔽₇) (order 2688) · Inn(G) = D₄×D₇ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

D₇×Q₁₆

Splittings

Fitting	C₇×Q₁₆ ⋊ C₂
Sylow	C₇ ⋊ C₂×Q₁₆

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 112);