Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2978 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #123×C53 | C4 |
| hypercenter | #123 | #6157 |
| hypocenter | C53 | #2978 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #123 | |
402 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 144 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,f^2=g,b^a=be,c^a=cg,d^a=de,h^a=h^30,c^b=ceg,d^b=dg,h^c=h^52,h^f=h^52220 conjugacy classes — view the full character table.
| maximal subgroup of | #28311539, #28311699, D53⋊#2978, … |
| maximal normal subgroup of | #28311539, #28311699, D53⋊#2978, … |
| maximal quotient of | #28318683, #28318693, #28318736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21388);