Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of (((C3⋊D3)⋊C3)⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (((C3⋊D3)⋊C3)⋊D3)⋊C3 |
| Fitting, Sylow | C32⋊H(3,𝔽3) ⋊ C22 |
| derived | C3×H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C22×C3 |
| Frattini, socle | C32 | C3×D32 |
| Fitting | C32⋊H(3,𝔽3) | C22 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 | |
226 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 18 |
Polycyclic presentation:
2^2,3^5|e^a=e^2,g^a=g^2,d^b=d^2,f^b=f^2,g^b=g^2,d^c=df,e^d=eg39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 428);