Finite Group DatabasebetaFinite Group Databasebeta| order | 5292 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C32×#64423 |
| Fitting, socle | C2×C32×C72 ⋊ C2×C3 |
| derived, Sylow | C72 ⋊ C22×C33 |
| Hall | C32×#3861 ⋊ C22 |
| Hall | C2×C7×D7 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | #64423 |
| commutator, cosocle, hypocenter | C72 | C22×C33 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C72 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C2×C32 | |
728 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
306 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C32×#64423⋊C2, C32×#11924207, C32×#11924176 |
| maximal normal subgroup of | C2×C32×#64423⋊C2, C32×#11924207, C32×#11924176 |
| maximal quotient of | C32×#11924207 |
Past the SmallGroups library coverage — no SmallGroups id.