Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C53 ⋊ #2616 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C22×#41×C53 | C2 |
| hypercenter | C22×#41 | D53 |
| hypocenter | C53 | #2616 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×#41 | |
816 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de,h^c=h^521072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27284404, #27284408, #27284432, … |
| maximal normal subgroup of | #27284404, #27284408, #27284432, … |
| maximal quotient of | #27310467, #27310516, #27310849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 10239);