Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C34 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×C3⋊D3 |
| commutator, Fitting, socle, cosocle | C2×C34 | C23 |
| hypocenter | C34 | #41 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
868 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 102 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:
2^4,3^4|a^2=d,c^2=d,b^a=bd,c^a=cd,h^a=h^2,c^b=cd,e^b=e^2,f^b=f^2,g^c=g^284 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3243);