C₂²×C₃⋊(C₃⋊D₃) #6339

order	216
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	Aut(G) order 7278336
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃³

Aliases

Fitting, socle	C₂²×C₃³ ⋊ C₂
derived, Sylow	C₃³ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂²

448 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);(3,4)(5,6,7);(1,2)(8,9,10);(11,12,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 176);