C₉⋊#157 #66420

order	324
isoclinism	isoclinic class of C₉⋊#157
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

C₃²×C₉ ⋊ C₄

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₃²×C₉	C₄
Frattini	C₃	C₃⋊#157
socle	C₃³	Dic₃
prime radical	C₃×C₃⋊D₃	D₃
cosocle	C₉×C₃⋊D₃	C₂
LCS	G ▷ C₃²×C₉
UCS	C₁

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 112);