Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #1740 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22⋊C4×D53 |
| commutator | C22×C53 | C2×C42 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#342 | C2 |
| socle | C22×C53 | C2×C22⋊C4 |
| hypercenter | #342 | D53 |
| hypocenter | C53 | #1740 |
| LCS | G ▷ C22×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #342 | |
420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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,b^2=f,c^2=g,b^a=bd,d^a=dg,e^b=eg,h^c=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27294265, #27294267, #27294275, … |
| maximal normal subgroup of | #27294265, #27294267, #27294275, … |
| maximal quotient of | #27273681, #27273687, #27273693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4241);