#13154994

order	1728
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

C₃³ ⋊ #339

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃²⋊D₃
commutator	C₃×C₃⋊Dic₃	C₄²
Fitting	C₂×C₃³×C₄	D₄
socle	C₂²×C₃³	C₂²⋊C₄
prime radical	C₂²×C₃×C₃⋊D₃	D₄
cosocle	C₂×C₃×C₄×C₃⋊D₃	C₂²
hypocenter	C₃³	#339
LCS	G ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄

416 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

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

μ_t(G) = 192

Polycyclic presentation:

2^6,3^3|a^2=e,b^2=d,c^2=f,d^2=f,b^a=bc,c^a=cf,g^a=g^2,h^a=h^2,i^a=h^2i,c^b=cf,h^b=h^2i^2,h^c=h^2,i^c=i^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 21057);