Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #2053 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133204 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#575 | C2 |
| socle | C2×C53 | #394 |
| hypercenter | #575 | D53 |
| hypocenter | C53 | #2053 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #575 | |
264 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 67 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,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^b=fg,h^c=h^52598 conjugacy classes — view the full character table.
| maximal subgroup of | #27296599, #27296604, #27296609, … |
| maximal normal subgroup of | #27296599, #27296604, #27296609, … |
| maximal quotient of | #27287823, #27287830, #27287837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5813);