C₂²⋊Dic₇ #1145

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) = C₂²

Aliases

C₂²⋊Dic₇

C₇ ⋊ C₂²⋊C₄

34 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 18);