Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#338 ⋊ C2 |
| Sylow | C53 ⋊ #2050 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133205 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#338 | C2 |
| socle | C2×C53 | #394 |
| hypercenter | #338 | D53 |
| hypocenter | C53 | #2050 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #338 | |
266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fg,h^a=h^52,c^b=cf,d^b=dg,e^b=eg,h^b=h^52,e^c=eg,f^c=fg,h^c=h^52598 conjugacy classes — view the full character table.
| maximal subgroup of | #27296619, #27296624, #27296630, … |
| maximal normal subgroup of | #27296619, #27296624, #27296630, … |
| maximal quotient of | #27287773, #27287780, #27287787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5798);