C₄⋊Dic₇ #1139

order	112
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) = #120×AGL(1,𝔽₇) (order 1344) · Inn(G) = C₂×D₇ · Out(G) = C₂×C₃×D₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₄⋊Dic₇

C₇ ⋊ C₄⋊C₄

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 112

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

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

Polycyclic presentation:

2^4,7|a^2=c,b^2=d,b^a=bd,e^a=e^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 12);