Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #131×C53 ⋊ C4 |
| Sylow | C53 ⋊ #2912 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C4⋊#6157 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C4 | C23×#6157 |
| Fitting | #131×C53 | C4 |
| socle | C2×C53 | C22×C4⋊C4 |
| hypercenter | #131 | #6157 |
| hypocenter | C53 | #2912 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
476 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 200 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=e,e^2=g,b^a=be,e^a=eg,h^a=h^30,d^c=dg,h^f=h^52304 conjugacy classes — view the full character table.
| maximal subgroup of | #28311494, #28311495, #28311636, … |
| maximal normal subgroup of | #28311494, #28311495, #28311636, … |
| maximal quotient of | #28316091, #28316092, #28316093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21164);