C₃⋊(C₃⋊(SL(2,𝔽₃)⋊C₈)) #13157018

order	1728
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃³

Aliases

unambiguous semidirect product	C₃⋊(C₃⋊(SL(2,𝔽₃)⋊C₈))
unambiguous semidirect product	SL(2,𝔽₃)⋊(C₃⋊(C₃⋊C₈))

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	C₃⋊(C₃⋊S₄)
commutator, hypocenter	C₃²×SL(2,𝔽₃)	C₈
Fitting	C₃²×C₄×Q₈	D₃
socle	C₂²×C₃²	SL(2,ℤ/4)
prime radical	C₂×C₃²×SL(2,𝔽₃)	C₄
cosocle	C₃²×C₄×SL(2,𝔽₃)	C₂
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄

492 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

μ_t(G) = 576

Polycyclic presentation:

2^3,3^3,2^3|a^2=b,b^2=c,g^2=i,h^2=i,d^a=d^2,e^a=e^2,f^a=f^2,g^a=gh,h^a=hi,g^d=h,h^d=ghi,h^g=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 23081);