D₁₁₂ #6373

order	224
isoclinism	isoclinic class of D₁₁₂
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(1,ℤ/16) (order 5376) · Inn(G) = D₅₆ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₁₁₂

Fitting	C₇×C₁₆ ⋊ C₂
Sylow	C₇ ⋊ D₁₆

28 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:

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

Polycyclic presentation:

2^5,7|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 5);