Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C83 ⋊ #339 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D332 |
| commutator | C4×C83 | C42 |
| Frattini | C22×C4 | C2×D83 |
| Fitting | C2×C42×C83 | C2 |
| socle | C22×C83 | C22⋊C4 |
| hypercenter | C2×C42 | D83 |
| hypocenter | C83 | #339 |
| LCS | G ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 | |
110 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=d,b^2=e,c^2=f,b^a=bc,c^a=cf,e^a=ef,g^a=g^821340 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #339×D83, C2×#19127461, #24907171, … |
| maximal normal subgroup of | #339×D83, C2×#19127461, #24907171, … |
| maximal quotient of | C2×#19127461 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 309);