C₃⋊AGL(1,𝔽₉) #6318

order	216
isoclinism	isoclinic class of C₃⋊AGL(1,𝔽₉)
automorphism	Aut(G) = D₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

C₃³ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³	C₈
prime radical	C₃×C₃⋊D₃	C₄
cosocle	C₃×#157	C₂
LCS	G ▷ C₃³
UCS	C₁

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

(2,3)(5,7,8,10,9,11,12,6);(1,2,3)(4,12,8)(5,11,6)(7,9,10)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 155);