Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×#1020⋊A4 |
| Fitting, socle | C27×C3 ⋊ C3×D3 |
| derived | C22×C22⋊A4 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #1020⋊A4 |
| commutator, cosocle, hypocenter | C22×C22⋊A4 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C27×C3 | C3×D3 |
| LCS | G ▷ C22×C22⋊A4 | |
| UCS | C1 ◁ C2×C3 | |
8876 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 8876 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#72215854 |
| maximal quotient of | C2×C3×#72215845 |
Past the SmallGroups library coverage — no SmallGroups id.