Dic₃⋊Dic₃ #3723

order	144
isoclinism	isoclinic class of #625
automorphism	Aut(G) = C₂⁴×D₃² (order 576) · Inn(G) = D₃² · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₃² ⋊ C₄⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃²
commutator	C₂×C₃²	C₂×C₄
Fitting, socle, prime radical, cosocle	C₂²×C₃²	C₂²
hypocenter	C₃²	C₄⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

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

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

Polycyclic presentation:

2^4,3^2|a^2=d,b^2=c,b^a=bd,f^a=f^2,e^b=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 66);