C₃⋊(#126⋊C₃) #63758

order	288
isoclinism	isoclinic class of C₃⋊(#126⋊C₃)
automorphism	Aut(G) = D₃×#560⋊C₃ (order 1152) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₃⋊(#126⋊C₃)
unambiguous semidirect product	#876⋊C₃

C₃×C₄² ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₄²	C₂×C₃
Frattini	C₂²	D₃×A₄
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ C₃×C₄²
UCS	C₁

43 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

(2,3)(5,16,18,9,10,14)(6,12,19,13,11,17)(7,8,15);(1,2,3)(4,17,15,10)(5,14,9,13)(6,12,18,11)(7,19,8,16)

Polycyclic presentation:

2,3,2^2,3,2^2|c^2=g,d^2=f,c^a=cfg,d^a=dg,e^a=e^2,c^b=cd,d^b=cf,f^b=g,g^b=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 406);