Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | Q8×#11719917 |
| Sylow | C53 ⋊ Q8×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C2×C4×Q8×C53 | C2 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | Q8×M16 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 | |
388 conjugacy classes of subgroups; only the 241 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 241 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1090 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27278425, #27278429, #27278439, … |
| maximal normal subgroup of | #27278425, #27278429, #27278439, … |
| maximal quotient of | C4⋊C4×#11719917, #27382744, Q8×#13133128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13464);