Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊#88622 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((C3⋊D3)⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C33×Q8 | C2×C3 |
| Frattini | C2×C3 | A4×C3⋊D3 |
| socle | C2×C32 | D3×A4 |
| LCS | G ▷ C33×Q8 | |
| UCS | C1 ◁ C2 | |
315 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3,2^2,3^3,2|a^2=h,c^2=h,d^2=h,e^a=e^2,f^a=f^2,g^a=g^2,c^b=cd,d^b=c,f^b=fg,d^c=dh57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2191);