Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, derived, Sylow | C32⋊H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ((C3⋊D3)⋊C3)⋊D3 |
| commutator, Fitting, cosocle, hypocenter | C32⋊H(3,𝔽3) | C22 |
| Frattini, socle | C32 | D3×C3⋊D3 |
| LCS | G ▷ C32⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
304 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 28 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,c^b=c^2,d^b=d^2,g^b=g^2,d^c=df,e^c=eg39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 455);