D₃² #158

order	36
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = GO⁺(4,𝔽₂) (order 72) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72213220
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

omega	Ω⁺(4,𝔽₂)
direct product	D₃²
unambiguous semidirect product	(C₃×D₃)⋊C₂
unambiguous semidirect product	(C₃⋊D₃)⋊C₂

C₃² ⋊ C₂²

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²	C₂²
LCS	G ▷ C₃²
UCS	C₁

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 6
transitive degree	μ_t(G) = 6

A minimal faithful permutation representation — 2 generators on 6 points:

(1,5,4,2,3,6);(1,5,3,2,4,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 10);