Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q32×C83 ⋊ C2 |
| Sylow | C83 ⋊ #510 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×D83 |
| commutator | C8×C83 | C23 |
| Frattini | C8 | C22×D83 |
| Fitting | Q32×C83 | C2 |
| socle | C2×C83 | C2×D8 |
| hypercenter | Q32 | D83 |
| hypocenter | C83 | #510 |
| LCS | G ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
82 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|b^2=f,d^2=ef,e^2=f,b^a=bd,c^a=cf,d^a=de,e^a=ef,g^a=g^82,d^b=de,e^b=ef,g^c=g^82467 conjugacy classes — view the full character table.
| maximal subgroup of | #510×D83, C2×#19128143, #24920596, … |
| maximal normal subgroup of | #510×D83, C2×#19128143, #24920596, … |
| maximal quotient of | #24909174, #24909181, #24909248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 991);