Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C53×#384 ⋊ C2 |
| Sylow | C53 ⋊ C4×#127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#384 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #384 | D53 |
| hypocenter | C53 | C4×#127 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #384 | |
924 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 249 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=e,c^2=f,d^2=g,b^a=be,c^a=cf,h^d=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | #27275031, #27275044, #27275057, … |
| maximal normal subgroup of | #27275031, #27275044, #27275057, … |
| maximal quotient of | #27305275, #384×Dic53, #27305879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7072);