C₃⋊(A₄⋊C₈) #63754

order	288
isoclinism	isoclinic class of C₃⋊S₄
automorphism	Aut(G) = C₂²×((C₃⋊S₄)⋊C₃)⋊C₂ (order 1728) · Inn(G) = C₃⋊S₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

unambiguous semidirect product	C₃⋊(A₄⋊C₈)
unambiguous semidirect product	A₄⋊(C₃⋊C₈)

C₃×A₄ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₃⋊S₄
commutator, hypocenter	C₃×A₄	C₈
Fitting	C₂²×C₃×C₄	D₃
socle	C₂³×C₃	Dic₃
prime radical	C₂×C₃×A₄	C₄
cosocle	C₃×C₄×A₄	C₂
LCS	G ▷ C₃×A₄
UCS	C₁ ◁ C₄

68 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 402);