Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#428 ⋊ C2 |
| Sylow | C53 ⋊ #2947 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#428 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #428 | D53 |
| hypocenter | C53 | #2947 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #428 | |
470 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=f,b^2=g,c^2=f,e^2=g,b^a=be,e^a=eg,e^b=eg,h^b=h^52,d^c=dg,h^c=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27404882, #27404884, #27404900, … |
| maximal normal subgroup of | #27404882, #27404884, #27404900, … |
| maximal quotient of | #27353539, #27353730, #27353751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13351);