D₅₆ #1133

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) = C₂

Aliases

D₅₆

Fitting	C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ D₈

22 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:

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

Polycyclic presentation:

2^4,7|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, 6);