C₇×C₈ #283

order	56
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₃ (order 24) · Inn(G) = C₁ · Out(G) = C₂³×C₃
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/8)×AGL(1,𝔽₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₆
abelian	C₇×C₈

8 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 1 generator on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 2);