Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#87588 |
| Fitting, socle | C27×C3 ⋊ D3 |
| derived | C22⋊(C22⋊A4) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #87588 |
| commutator, cosocle, hypocenter | C22⋊(C22⋊A4) | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C27×C3 | D3 |
| LCS | G ▷ C22⋊(C22⋊A4) | |
| UCS | C1 ◁ C2×C3 | |
13778 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 13778 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#87588⋊C2, C2×D3×#87588, C2×#72231611, … |
| maximal normal subgroup of | C2×C3×#87588⋊C2, C2×D3×#87588, C2×#72231611, … |
| maximal quotient of | C2×C3×#11715850 |
Past the SmallGroups library coverage — no SmallGroups id.