Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊(#41⋊H(3,𝔽3)) |
| unambiguous semidirect product | #239⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C3⋊(#41⋊C3))⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C3 | C3×D3×A4 |
| commutator, cosocle | C32×Q8 | C2×C32 |
| Fitting | C33×Q8 | C2×C3 |
| socle | C2×C32 | C2×C3×A4 |
| hypercenter | C2×C32 | D3×A4 |
| hypocenter | C3×Q8 | C2×H(3,𝔽3) |
| LCS | G ▷ C32×Q8 ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 | |
229 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 36 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:
2,3^3,2^2,3,2|a^2=h,e^2=h,f^2=h,g^a=g^2,c^b=cd,e^b=ef,f^b=e,f^e=fh93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2167);