Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1953 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133191 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#405 | C2 |
| socle | C2×C53 | #389 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #1953 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 | |
184 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 89 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=f,b^2=f,c^2=e,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^52,d^b=dg1072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27300733, #27300736, #27300768, … |
| maximal normal subgroup of | #27300733, #27300736, #27300768, … |
| maximal quotient of | #27282703, #27282708, #27282713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5300);