C₃⋊(C₃⋊Dic₃) #1107

order	108
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	Aut(G) order 606528
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃³

Aliases

C₃³ ⋊ C₄

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

84 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 34);