C₄²⋊C₉ #3660

order	144
isoclinism	isoclinic class of C₄²⋊C₃
automorphism	Aut(G) = C₃×#4626⋊C₂ (order 1152) · Inn(G) = C₄²⋊C₃ · Out(G) = C₃×D₄
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₄

Aliases

C₄²⋊C₉

C₄² ⋊ C₉

	subgroup	G/H
center, hypercenter	C₃	C₄²⋊C₃
commutator, hypocenter	C₄²	C₉
Frattini, socle, prime radical	C₂²×C₃	A₄
Fitting, cosocle	C₃×C₄²	C₃
LCS	G ▷ C₄²
UCS	C₁ ◁ C₃

17 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 3);