Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C53×#390 ⋊ C2 |
| Sylow | C53 ⋊ #2363 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#390 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #390 | D53 |
| hypocenter | C53 | #2363 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #390 | |
688 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=eg,c^2=e,d^2=e,b^a=be,c^a=cf,h^a=h^52,d^b=de,h^b=h^52,h^c=h^52778 conjugacy classes — view the full character table.
| maximal subgroup of | #27404393, #27404400, #27404438, … |
| maximal normal subgroup of | #27404393, #27404400, #27404438, … |
| maximal quotient of | #27305516, #27305619, #27305639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7553);