Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | D32×#1020⋊C2 |
| Fitting | C32×#120 ⋊ C22×D3 |
| derived | C32×C22⋊A4 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C22⋊A4 | C24 |
| Frattini | C22 | C2×D32×S4 |
| Fitting | C32×#120 | C22×D3 |
| socle | C22×C32 | C23×S4 |
| LCS | G ▷ C32×C22⋊A4 | |
| UCS | C1 | |
12741 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 12741 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
126 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72210357 |
| maximal quotient of | D3×#72210357 |
Past the SmallGroups library coverage — no SmallGroups id.