Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#72213207 |
| derived | (C22⋊A4)⋊C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72213207 |
| commutator, cosocle, hypocenter | (C22⋊A4)⋊C9 | C22×C3 |
| Frattini | C3 | C2×C3×A4⋊S4 |
| Fitting | C25×C32 | C3⋊D3 |
| socle | C25×C3 | C3×C3⋊D3 |
| LCS | G ▷ (C22⋊A4)⋊C9 | |
| UCS | C1 ◁ C2 | |
1314 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1314 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.
100 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72213207, C4×#72213207 |
| maximal normal subgroup of | C22×#72213207, C4×#72213207 |
| maximal quotient of | C2×#72220160, C2×#72219864, #72245689, … |
| holomorph of | C2×C22⋊C9 |
Past the SmallGroups library coverage — no SmallGroups id.