Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2374 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C2×#123×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#123 | D53 |
| hypocenter | C53 | #2374 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
552 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=g,b^2=g,c^2=ef,d^2=ef,b^a=be,c^a=cf,h^a=h^52,d^b=de,h^b=h^52778 conjugacy classes — view the full character table.
| maximal subgroup of | #27300850, #27300854, #27301357, … |
| maximal normal subgroup of | #27300850, #27300854, #27301357, … |
| maximal quotient of | #27305558, #27305652, #27305679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7646);