Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #3407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×#6157 |
| commutator, socle | C2×C53 | C23×C8 |
| Fitting | #131×C53 | C4 |
| hypercenter | #131 | #6157 |
| hypocenter | C53 | #3407 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ #131 | |
360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 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=e,e^2=f,f^2=g,d^a=dg,h^a=h^30,c^b=cg,h^d=h^52,h^e=h^52328 conjugacy classes — view the full character table.
| maximal subgroup of | #28310574, #28310575, #28311732, … |
| maximal normal subgroup of | #28310574, #28310575, #28311732, … |
| maximal quotient of | #28327903, #28327914, #28327915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21478);