Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | #127×GL(2,𝔽3) |
| Fitting | Q8×#127 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C23×S4 |
| commutator, cosocle | C22×SL(2,𝔽3) | C24 |
| Fitting | Q8×#127 | D3 |
| hypercenter | C2×#127 | S4 |
| hypocenter | SL(2,𝔽3) | C2×#127 |
| LCS | G ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
4939 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 4939 conjugacy classes of subgroups — too many to draw. Only its 236 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
112 conjugacy classes — view the full character table.
| maximal subgroup of | #127×AGL(2,𝔽3) |
| maximal quotient of | #127×AGL(2,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id.