Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#383 ⋊ C2 |
| Sylow | C53 ⋊ C4×#126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#383 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #383 | D53 |
| hypocenter | C53 | C4×#126 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #383 | |
516 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 189 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=ef,c^2=e,d^2=g,b^a=be,c^a=cf,h^d=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#126×D53, C2×#20277263, #27542586, … |
| maximal normal subgroup of | C4×#126×D53, C2×#20277263, #27542586, … |
| maximal quotient of | #27305245, #27305264, #383×Dic53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7058);