Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of #4207⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | #66409⋊C3 |
| derived | C3×C9 ⋊ C32×C4 |
| Sylow | 3−1+4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4207⋊C3 |
| commutator, hypocenter | C3×C9 | C32×C4 |
| Frattini, socle | C2×C3 | C32×C3⋊D3 |
| Fitting | C2×3−1+4 | C2 |
| prime radical | C2×C3×H(3,𝔽3) | D3 |
| cosocle | C2×C3×C9 | C2×C32 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2 | |
318 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3^2,2,3^3|a^2=d,e^3=g,e^a=e^2g^2,f^a=f^2,g^a=g^2,f^b=fg^2,e^c=eg^2110 conjugacy classes — view the full character table.
| maximal subgroup of | #10619905⋊C3, (#4207⋊C3)⋊C4, #10619908⋊C3, … |
| maximal normal subgroup of | #10619905⋊C3, (#4207⋊C3)⋊C4, #10619908⋊C3, … |
| maximal quotient of | #10619724⋊C3, C2×#66409⋊C3, #72247228 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 739);