Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C53×#585 ⋊ C2 |
| Sylow | C53 ⋊ 2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D53 |
| commutator, socle | C2×C53 | C26 |
| Fitting | C53×#585 | C2 |
| hypercenter | #585 | D53 |
| hypocenter | C53 | 2−1+6 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ #585 | |
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^e=h^52,h^f=h^52949 conjugacy classes — view the full character table.
| maximal subgroup of | #27532840, #27532842, #27532844, … |
| maximal normal subgroup of | #27532840, #27532842, #27532844, … |
| maximal quotient of | #28290712, #28290714, #28290716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19323);