Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | #123×D83 |
| Fitting | #123×C83 ⋊ C2 |
| Sylow | C83 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D83 |
| commutator, socle | C22×C83 | C24 |
| Fitting | #123×C83 | C2 |
| hypercenter | #123 | D83 |
| hypocenter | C83 | C2×#123 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ #123 | |
342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
602 conjugacy classes — view the full character table.
| maximal subgroup of | #352×D83, #458×D83, C2×#123×D83, … |
| maximal normal subgroup of | #352×D83, #458×D83, C2×#123×D83, … |
| maximal quotient of | #381×D83, #385×D83, #386×D83, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1049);