C₄⋊(C₇⋊C₈) #6378

order	224
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) = C₂×#120×AGL(1,𝔽₇) (order 2688) · Inn(G) = C₂×D₇ · Out(G) = C₂²×C₃×D₄
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₄⋊(C₇⋊C₈)

C₇ ⋊ C₄⋊C₈

38 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2^5,7|a^2=c,b^2=d,c^2=e,b^a=bd,f^a=f^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 10);