C₃⋊#622 #6281

order	216
isoclinism	isoclinic class of C₃⋊#157
automorphism	Aut(G) = C₂×D₃×AΓL(1,𝔽₉) (order 1728) · Inn(G) = C₃⋊#157 · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

C₃³ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊#157
commutator, hypocenter	C₃³	C₈
Fitting, socle, prime radical	C₂×C₃³	C₄
cosocle	C₃×C₃⋊Dic₃	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 118);