Dic₂₈ #1134

order	112
isoclinism	isoclinic class of #1132
automorphism	Aut(G) = AGL(1,ℤ/8)×AGL(1,𝔽₇) (order 1344) · Inn(G) = D₂₈ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₂₈

C₇ ⋊ Q₁₆

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 7);