Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | M16×#88322 |
| Sylow | C53 ⋊ C8×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D53 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | C4×M16×C53 | C2 |
| socle | C22×C53 | C2×C42 |
| hypercenter | C4×M16 | D53 |
| hypocenter | C53 | C8×M16 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C4×M16 | |
224 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 153 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 | #27274649, #100×#88322, #27274654, … |
| maximal normal subgroup of | #27274649, #100×#88322, #27274654, … |
| maximal quotient of | #97×#88322, #27273473, #27274517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2788);