Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C22×C42×C53 ⋊ C2 |
| Sylow | C53 ⋊ #3444 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D53 |
| commutator, socle | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C22×C42×C53 | C2 |
| hypercenter | C22×C42 | D53 |
| hypocenter | C53 | #3444 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|d^2=g,e^2=f,b^a=bf,c^a=cg,h^a=h^521720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27350603, #27350606, #27350626, … |
| maximal normal subgroup of | #27350603, #27350606, #27350626, … |
| maximal quotient of | #27542394, #27542500, C4×#16811134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17806);