C₂²⋊(C₂²⋊C₉) #3768

order	144
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	Aut(G) = C₃×AΓL(2,𝔽₄) (order 17280) · Inn(G) = C₂²⋊A₄ · Out(G) = C₃×S₅
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂²⋊(C₂²⋊C₉)

C₂⁴ ⋊ C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₉
Fitting, socle, prime radical, cosocle	C₂⁴×C₃	C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃

61 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 111);