Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2684 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C2×#120×C53 | C2 |
| hypercenter | C2×#120 | D53 |
| hypocenter | C53 | #2684 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
688 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|b^2=f,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | #27303063, #27303065, #27303539, … |
| maximal normal subgroup of | #27303063, #27303065, #27303539, … |
| maximal quotient of | #27311379, #27311484, #27311890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 10888);