#8135⋊C₃ #11709121

order	768
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, hypercenter	C₂	#1454⋊C₃
commutator, cosocle, hypocenter	#1273	C₂×C₃
Frattini	C₂×C₄²	C₂×A₄
Fitting	#8135	C₃
socle	C₂³	C₂×C₄²⋊C₃
prime radical	#1273⋊C₃	C₂
LCS	G ▷ #1273
UCS	C₁ ◁ C₂

82 conjugacy classes of subgroups.

μ_t(G) = 48

Polycyclic presentation:

3,2^8|b^2=f,c^2=g,d^2=e,f^2=h,g^2=i,b^a=cfhi,c^a=bcdgh,d^a=df,e^a=eh,f^a=gh,g^a=fg,h^a=i,i^a=hi,c^b=ce,d^b=dhi,d^c=dh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1083502);