D₃×D₉ #1089

order	108
isoclinism	isoclinic class of D₃×D₉
automorphism	Aut(G) = D₃×AGL(1,ℤ/9) (order 324) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×D₉
unambiguous semidirect product	(C₃×D₉)⋊C₂
unambiguous semidirect product	(D₃×C₉)⋊C₂
unambiguous semidirect product	(C₃⋊D₉)⋊C₂

C₃×C₉ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₉	C₂²
Frattini	C₃	D₃²
socle	C₃²	C₂×D₃
LCS	G ▷ C₃×C₉
UCS	C₁

34 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 16);