Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C83 ⋊ #419 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C22⋊D83 |
| commutator | C4×C83 | C22×C4 |
| Frattini | C2×C4 | C22×D83 |
| Fitting | #117×C83 | C2 |
| socle | C22×C83 | C2×D4 |
| hypercenter | #117 | D83 |
| hypocenter | C83 | #419 |
| LCS | G ▷ C4×C83 ▷ C2×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #117 | |
150 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,83|a^2=e,b^2=f,c^2=f,d^2=f,b^a=bd,d^a=df,c^b=cf,d^b=df,g^b=g^82842 conjugacy classes — view the full character table.
| maximal subgroup of | #24907436, #24907437, #24907832, … |
| maximal normal subgroup of | #24907436, #24907437, #24907832, … |
| maximal quotient of | #24907184, #24907296, #24907397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 557);