C₃⋊(C₃⋊C₁₆) #3686

order	144
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×C₄×AGL(2,𝔽₃) (order 3456) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×C₄×S₄
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₃

Aliases

C₃⋊(C₃⋊C₁₆)

C₃² ⋊ C₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₈	C₃⋊D₃
commutator, hypocenter	C₃²	C₁₆
Fitting, cosocle	C₃²×C₈	C₂
socle, prime radical	C₂×C₃²	C₈
LCS	G ▷ C₃²
UCS	C₁ ◁ C₈

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,19)(21,22);(20,21,22);(17,18,19)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 29);