Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C83 ⋊ #344 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | (C2×D83)⋊C4 |
| commutator | C22×C83 | C42 |
| Frattini | C22×C4 | C2×D83 |
| Fitting | C2×M16×C83 | C2 |
| socle | C2×C83 | #95 |
| hypercenter | C2×M16 | D83 |
| hypocenter | C83 | #344 |
| LCS | G ▷ C22×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×M16 | |
90 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=d,b^2=e,e^2=f,b^a=bc,c^a=cf,g^a=g^82,d^b=df,g^b=g^82842 conjugacy classes — view the full character table.
| maximal subgroup of | #344×D83, C2×#19127473, #24907264, … |
| maximal normal subgroup of | #344×D83, C2×#19127473, #24907264, … |
| maximal quotient of | C2×#19127473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 321);