C₃⋊(C₃⋊C₈) #616

order	72
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂²×AGL(2,𝔽₃) (order 1728) · Inn(G) = C₃⋊D₃ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

C₃⋊(C₃⋊C₈)

C₃² ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₃⋊D₃
commutator, hypocenter	C₃²	C₈
Fitting, cosocle	C₃²×C₄	C₂
socle, prime radical	C₂×C₃²	C₄
LCS	G ▷ C₃²
UCS	C₁ ◁ C₄

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 13);