Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C83⋊#564 |
| Sylow | C83 ⋊ #564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D83 |
| commutator, socle | C22×C83 | C24 |
| Fitting | #125×C83 | C2 |
| hypercenter | #125 | D83 |
| hypocenter | C83 | #564 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ #125 | |
172 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=e,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,g^d=g^82593 conjugacy classes — view the full character table.
| maximal subgroup of | #24906764, #24906766, #24906789, … |
| maximal normal subgroup of | #24906764, #24906766, #24906789, … |
| maximal quotient of | #24910645, #24910646, #24910647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1304);