(D₃²⋊D₃)⋊C₈ #13165173

order	1728
isoclinism	isoclinic class of D₃×GO⁺(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

(D₃²⋊D₃)⋊C₈

C₃³ ⋊ C₈×D₄

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×GO⁺(4,𝔽₂)
commutator	C₃×C₃⋊D₃	C₂²×C₈
Fitting	C₃³×C₄	C₂×D₄
socle	C₂×C₃³	C₄×D₄
prime radical	C₂×D₃²⋊D₃	C₄
cosocle	C₃×C₄×C₃⋊D₃	C₂³
hypocenter	C₃³	C₈×D₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₄

464 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

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

μ_t(G) = 96

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 31236);