C₃⋊SL(2,ℤ/4) #3783

order	144
isoclinism	isoclinic class of C₃⋊S₄
automorphism	Aut(G) = C₂×((C₃⋊S₄)⋊C₃)⋊C₂ (order 864) · Inn(G) = C₃⋊S₄ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

unambiguous semidirect product	C₃⋊SL(2,ℤ/4)
unambiguous semidirect product	A₄⋊Dic₃

C₃×A₄ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊S₄
commutator, hypocenter	C₃×A₄	C₄
Fitting, socle	C₂³×C₃	D₃
prime radical, cosocle	C₂×C₃×A₄	C₂
LCS	G ▷ C₃×A₄
UCS	C₁ ◁ C₂

50 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 11 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 126);