Dic₇⋊C₈ #6388

order	224
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) = C₂⁵×AGL(1,𝔽₇) (order 1344) · Inn(G) = C₂×D₇ · Out(G) = C₂⁴×C₃
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

Dic₇⋊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)(9,11)(12,13,14,16,15,17,18,19);(2,3)(4,5)(6,7)(8,9,10,11)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 20);