Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#340 ⋊ C2 |
| Sylow | C53 ⋊ #2067 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | D4×D53 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#340 | C2 |
| socle | C23×C53 | C2×D4 |
| hypercenter | #340 | D53 |
| hypocenter | C53 | #2067 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ #340 | |
284 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=de,b^2=dg,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^52,c^b=cf,h^b=h^52,d^c=dg,e^c=eg,f^c=fg,h^c=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2067, C2×#20276104, #27412704, … |
| maximal normal subgroup of | D53×#2067, C2×#20276104, #27412704, … |
| maximal quotient of | #27288610, #27288617, #27288638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5899);