Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C2×#72234214 |
| Fitting | C22×H(7,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#87524 |
| commutator, cosocle | #5950 | C24 |
| Frattini | C4 | C23×#1020 |
| Fitting | C22×H(7,𝔽2) | D3 |
| hypercenter | C22×D4 | #1020 |
| hypocenter | #997 | C22×D4 |
| LCS | G ▷ #5950 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×D4 | |
87165 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 87165 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
118 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72241002, C2×#72240994, C2×#72240979 |
| maximal normal subgroup of | C2×#72241002, C2×#72240994, C2×#72240979 |
Past the SmallGroups library coverage — no SmallGroups id.