#13165757

order	1728
isoclinism	isoclinic class of D₃×PSU(3,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

C₃³ ⋊ C₈×Q₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×PSU(3,𝔽₂)
commutator	C₃×C₃⋊D₃	C₂²×C₈
Fitting	C₃³×C₄	C₂×Q₈
socle	C₂×C₃³	C₄×Q₈
prime radical	C₂×C₃×C₃⋊D₃	C₂²×C₄
cosocle	C₃×C₄×C₃⋊D₃	C₂³
hypocenter	C₃³	C₈×Q₈
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₄

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

normal subgroups only

This group has 234 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) = 192

Polycyclic presentation:

2^6,3^3|a^2=d,b^2=d,c^2=ef,d^2=f,e^2=f,b^a=bdef,h^a=i,i^a=h^2,h^b=hi^2,i^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, 31820);