Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C83 ⋊ #390 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D83 |
| commutator | C22×C83 | C22×C4 |
| Fitting | C22×D4×C83 | C2 |
| socle | C23×C83 | C23 |
| hypercenter | C22×D4 | D83 |
| hypocenter | C83 | #390 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,83|a^2=f,b^2=df,c^2=e,b^a=bd,c^a=ce,g^a=g^82,g^b=g^82848 conjugacy classes — view the full character table.
| maximal subgroup of | #24907417, #24907419, #24907673, … |
| maximal normal subgroup of | #24907417, #24907419, #24907673, … |
| maximal quotient of | #24908258, #24908262, #24908266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 452);