Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C42 |
| direct product | #127×#127⋊C3 |
| Fitting, Sylow | #1272 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×#127⋊C3 |
| commutator, cosocle | C22×C42 | C24×C3 |
| Frattini, socle | C24 | C24×A4 |
| Fitting | #1272 | C3 |
| hypercenter | #127 | #127⋊C3 |
| hypocenter | C42 | C2×C3×#127 |
| LCS | G ▷ C22×C42 ▷ C42 | |
| UCS | C1 ◁ C22 ◁ #127 | |
19824 conjugacy classes of subgroups; only the 341 normal subgroups are shown.
This group has 19824 conjugacy classes of subgroups — too many to draw. Only its 341 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #127⋊C32, #72258954 |
| maximal normal subgroup of | #127⋊C32, #72258954 |
Past the SmallGroups library coverage — no SmallGroups id.