Dic₂₁ #746

order	84
isoclinism	isoclinic class of D₂₁
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₇) (order 504) · Inn(G) = D₂₁ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72222273
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₁
unambiguous semidirect product	C₃⋊Dic₇
unambiguous semidirect product	C₇⋊Dic₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₁
commutator, hypocenter	C₃×C₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 5);