C₃⋊#859 #63755

order	288
isoclinism	isoclinic class of C₃⋊#226
automorphism	Aut(G) = C₂²×((C₃⋊S₄)⋊C₃)⋊C₂ (order 1728) · Inn(G) = C₃⋊S₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂²	C₃⋊S₄
commutator, hypocenter	C₃×SL(2,𝔽₃)	C₄
Fitting	C₂×C₃×Q₈	D₃
socle	C₂²×C₃	S₄
prime radical, cosocle	C₂×C₃×SL(2,𝔽₃)	C₂
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

78 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 403);