Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×Q8×#88322 |
| Sylow | C53 ⋊ C2×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C2×C4×Q8×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | C2×C8×Q8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×Q8 | |
440 conjugacy classes of subgroups; only the 333 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 333 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #101×#88322, #27274658, #27274667, … |
| maximal normal subgroup of | #101×#88322, #27274658, #27274667, … |
| maximal quotient of | C2×C4⋊C4×#88322, C4×Q8×#88322, C2×#20275274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13432);