D₃²⋊C₄ #3772

order	144
isoclinism	isoclinic class of GO⁺(4,𝔽₂)
automorphism	Aut(G) = C₂²×GO⁺(4,𝔽₂) (order 288) · Inn(G) = GO⁺(4,𝔽₂) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

D₃²⋊C₄

C₃² ⋊ C₂²⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃²	D₄
prime radical	C₂×D₃²	C₂
cosocle	C₂×C₃⋊D₃	C₂²
hypocenter	C₃²	C₂²⋊C₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂

58 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 10 points:

(1,3)(2,5,6)(7,8,9,10);(1,2)(3,6)(4,5)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 115);