Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×#123×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2471 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C2×#123×C53 | C2 |
| hypercenter | C2×#123 | D53 |
| hypocenter | C53 | #2471 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
582 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,b^2=ef,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^52766 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2471, C2×#20278822, #27552679, … |
| maximal normal subgroup of | D53×#2471, C2×#20278822, #27552679, … |
| maximal quotient of | #27306232, #27306670, #27307051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 8617);