Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊(C83⋊C16) |
| Sylow | C83 ⋊ C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C2×D83 |
| commutator | C2×C83 | C2×C16 |
| Fitting | C22×C8×C83 | C2 |
| socle | C22×C83 | C2×C8 |
| hypercenter | C22×C8 | D83 |
| hypocenter | C83 | C22⋊C16 |
| LCS | G ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 | |
66 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=d,d^2=e,e^2=f,b^a=bc,g^a=g^821352 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22⋊C16×D83, C2×C22⋊(C83⋊C16), #24909072, … |
| maximal normal subgroup of | C22⋊C16×D83, C2×C22⋊(C83⋊C16), #24909072, … |
| maximal quotient of | C2×C22⋊(C83⋊C16) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 329);