C₃⋊#1113 #66478

order	324
isoclinism	isoclinic class of C₃⋊#1113
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

C₃⁴ ⋊ C₂²

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃⁴	C₂²
LCS	G ▷ C₃⁴
UCS	C₁

224 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 36

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 170);