Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2274 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C2×C53 | C22×C42 |
| Fitting | C4×C22⋊C4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C4×C22⋊C4 | D53 |
| hypocenter | C53 | #2274 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
696 conjugacy classes of subgroups; only the 325 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 325 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=f,b^2=g,c^2=e,b^a=be,d^a=de,c^b=ce,h^c=h^521108 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27277553, #27281529, #27281531, … |
| maximal normal subgroup of | #27277553, #27281529, #27281531, … |
| maximal quotient of | #27305174, #27305176, #27305335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6865);