((C₃⋊D₃)⋊C₉)⋊C₂ #66345

order	324
isoclinism	isoclinic class of ((C₃⋊D₃)⋊C₉)⋊C₂
automorphism	Aut(G) = (((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃ (order 972) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	((C₃⋊D₃)⋊C₉)⋊C₂
unambiguous semidirect product	#4169⋊C₂
unambiguous semidirect product	#4181⋊C₂

C₃²⋊C₉ ⋊ C₂²

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

67 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 18 points:

(3,9,11,8,10,17)(4,5)(6,18,16,13,15,14)(7,12);(1,15,8,12,10,14,4,6,17,2,11,13,5,16,9,7,3,18)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 37);