C₇⋊C₈ #282

order	56
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂²×AGL(1,𝔽₇) (order 168) · Inn(G) = D₇ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72214550
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

C₇⋊C₈

C₇ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₇
commutator, hypocenter	C₇	C₈
Fitting, cosocle	C₄×C₇	C₂
socle, prime radical	C₂×C₇	C₄
LCS	G ▷ C₇
UCS	C₁ ◁ C₄

8 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^3,7|a^2=b,b^2=c,d^a=d^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 1);