C₃⋊(#41⋊C₉) #88620

order	432
isoclinism	isoclinic class of C₃⋊(#41⋊C₃)
automorphism	Aut(G) = C₂×C₃×D₃×S₄ (order 864) · Inn(G) = D₃×A₄ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₃⋊(#41⋊C₉)
unambiguous semidirect product	#239⋊C₉

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	D₃×A₄
commutator, hypocenter	C₃×Q₈	C₂×C₉
Fitting, cosocle	C₃²×Q₈	C₂×C₃
socle	C₂×C₃²	C₂×A₄
prime radical	C₃×#239	C₃
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂×C₃

57 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,5,8)(3,13,12,14,9,7,10,15,4,6,11,16)(18,19)(20,24,28,22,27,23,25,21,26);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(17,18,19)

Polycyclic presentation:

2,3^2,2^2,3,2|a^2=g,b^3=c,d^2=g,e^2=g,f^a=f^2,d^b=de,e^b=d,e^d=eg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 267);