Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C32⋊C9 ⋊ C2 |
| Sylow | C32⋊C9 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×C3⋊D9 |
| commutator, cosocle | C2×C32⋊C9 | C23 |
| Frattini, socle | C2×C32 | C22×C3⋊D3 |
| Fitting | Q8×C32⋊C9 | C2 |
| hypercenter | C3×Q8 | C3⋊D9 |
| hypocenter | C32⋊C9 | #41 |
| LCS | G ▷ C2×C32⋊C9 ▷ C32⋊C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
392 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 83 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|b^2=d,c^2=d,f^3=h,c^a=cd,e^a=e^2,f^a=f^2h^2,h^a=h^2,c^b=cd,f^e=fg105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1176);