Q₈×D₇ #1160

order	112
automorphism	Aut(G) = C₂×S₄×AGL(1,𝔽₇) (order 2016) · Inn(G) = C₂²×D₇ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×D₇

Fitting	C₇×Q₈ ⋊ C₂
Sylow	C₇ ⋊ C₂×Q₈

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 33);