Dic₄₂ #4264

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) = C₁

Aliases

Dic₄₂

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₇	C₂²
Fitting	C₃×C₄×C₇	C₂
hypercenter	C₄	D₂₁
hypocenter	C₃×C₇	Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2^3,3,7|a^2=c,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, 34);