C₃⋊#1088 #66421

order	324
isoclinism	isoclinic class of C₃⋊#1088
automorphism	Aut(G) = D₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = C₃⋊#157 · Out(G) = D₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊#157
commutator, Fitting, hypocenter	C₃×H(3,𝔽₃)	C₄
socle	C₃²	#157
prime radical, cosocle	C₃×#272	C₂
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃

50 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 113);