C₃⋊#857 #63753

order	288
isoclinism	isoclinic class of C₃⋊#857
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃

Aliases

Fitting	C₃×C₄² ⋊ D₃
derived	C₃×C₄²⋊C₃ ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₄²⋊C₃	C₂
Frattini	C₂²	C₃⋊S₄
Fitting	C₃×C₄²	D₃
socle	C₂²×C₃	S₄
LCS	G ▷ C₃×C₄²⋊C₃
UCS	C₁

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 36

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

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

Polycyclic presentation:

2,3^2,2^4|d^2=fg,e^2=f,b^a=b^2,c^a=c^2,d^a=e,e^a=d,f^a=fg,d^b=e,e^b=deg,f^b=g,g^b=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 401);