Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11724052 |
| commutator, cosocle, hypocenter | C3×#997 | C2×C3 |
| Frattini, socle | C2×C3 | C3×#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| prime radical | C32⋊#997 | C2 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2 | |
315 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,c^a=c^2,d^a=d^2,e^a=ef,g^a=gh,c^b=cd^2,e^c=f,f^c=ef,g^c=h,h^c=gh,h^e=hi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30596);