Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10619775 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived | C22×C33 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10619775 |
| commutator, hypocenter | C22×C33 | C3×C4 |
| Frattini | C2×C32 | D3×A4 |
| Fitting, cosocle | C23×C33 | C2×C3 |
| socle | C23×C3 | (C3⋊D3)⋊C3 |
| prime radical | C2×#66368 | C2 |
| LCS | G ▷ C22×C33 | |
| UCS | C1 ◁ C2 | |
300 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3,2^3,3^3|a^2=c,f^a=f^2,g^a=g^2,h^a=h^2,d^b=de,e^b=d,f^b=fg,g^b=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 783);