Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C4⋊#88322 |
| Sylow | C53 ⋊ C4×C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D53 |
| commutator | C2×C53 | C2×C4×C8 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C43×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C43 | D53 |
| hypocenter | C53 | C4×C4⋊C8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C42 ◁ C43 | |
316 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1744 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27270619, #27270620, #27270634, … |
| maximal normal subgroup of | #27270619, #27270620, #27270634, … |
| maximal quotient of | C8×C4⋊#88322, #27275217, #27275226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4359);