Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×#13133139 |
| Fitting | C23×C8×C53 ⋊ C2 |
| Sylow | C53 ⋊ C22×#102 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | D212 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C23×C8×C53 | C2 |
| socle | C24×C53 | D4 |
| hypercenter | C23×C8 | D53 |
| hypocenter | C53 | C22×#102 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×C4 ◁ C23×C8 | |
1072 conjugacy classes of subgroups; only the 349 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 349 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27289655, #27289659, #27290353, … |
| maximal normal subgroup of | #27289655, #27289659, #27290353, … |
| maximal quotient of | C22×#16810352, C2×#20273141, C2×#20273115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 12997);