Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C22×C42×C53 ⋊ C2 |
| Sylow | C53 ⋊ #1437 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D53 |
| commutator | C22×C53 | C2×C42 |
| Fitting | C22×C42×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C42 | D53 |
| hypocenter | C53 | #1437 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C42 | |
828 conjugacy classes of subgroups; only the 241 normal subgroups are shown.
This group has 828 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.
Polycyclic presentation:
2^7,53|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,h^a=h^521720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27275604, D53×#1437, C2×#20272953, … |
| maximal normal subgroup of | #27275604, D53×#1437, C2×#20272953, … |
| maximal quotient of | #27272012, #27272013, #27272015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2748);