D₈₄ #4266

order	168
isoclinism	isoclinic class of Dic₄₂
automorphism	Aut(G) = D₃×D₄×AGL(1,𝔽₇) (order 2016) · Inn(G) = C₂×D₂₁ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₄
unambiguous semidirect product	C₄⋊D₂₁

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Polycyclic presentation:

2^3,3,7|b^2=c,b^a=bc,d^a=d^2,e^a=e^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 36);