C₃⋊D₉ #271

order	54
isoclinism	isoclinic class of C₃⋊D₉
automorphism	Aut(G) = #72210336 (order 2916) · Inn(G) = G · Out(G) = #272
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊D₉
unambiguous semidirect product	C₉⋊D₃

C₃×C₉ ⋊ C₂

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

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 3 generators on 12 points:

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

Polycyclic presentation:

2,3^3|b^3=d^2,b^a=b^2d,c^a=c^2,d^a=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 7);