(C₃⋊D₃)⋊A₄ #6262

order	216
isoclinism	isoclinic class of (C₃⋊D₃)⋊A₄
automorphism	Aut(G) = ((C₃⋊D₃)⋊C₃)⋊S₄ (order 1296) · Inn(G) = G · Out(G) = D₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

(C₃⋊D₃)⋊A₄

C₂²×C₃² ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂²×C₃²	C₂×C₃
Frattini	C₃	D₃×A₄
socle	C₂²×C₃	C₃×D₃
LCS	G ▷ C₂²×C₃²
UCS	C₁

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 99);