Finite Group DatabasebetaFinite Group Databasebeta| order | 12168 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#72210302 |
| Fitting, socle | C3×C132 ⋊ C2×C3×C4 |
| derived, Sylow | C132 ⋊ C2×C32×C4 |
| Hall | C3×#92788 ⋊ C2×C4 |
| Hall | D13⋊#261 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72210302 |
| commutator, hypocenter | C132 | C2×C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C132 | C2×C3×C4 |
| prime radical | C3×#11759815⋊C2 | C2 |
| cosocle | C13⋊D13 | C22×C32 |
| LCS | G ▷ C132 | |
| UCS | C1 ◁ C3 | |
300 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72210302, C3×#72214122, #72242001 |
| maximal normal subgroup of | D3×#72210302, C3×#72214122, #72242001 |
Past the SmallGroups library coverage — no SmallGroups id.