Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#392 ⋊ C2 |
| Sylow | C53 ⋊ #2397 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#392 | C2 |
| hypercenter | #392 | D53 |
| hypocenter | C53 | #2397 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #392 | |
840 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 167 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=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^d=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27294418, #27294855, #27356415, … |
| maximal normal subgroup of | #27294418, #27294855, #27356415, … |
| maximal quotient of | #27305825, #27305827, #27305872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7788);