Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #3113 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#432 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #432 | D53 |
| hypocenter | C53 | #3113 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #432 | |
380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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=g,c^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^52,d^c=dg,f^c=fg,h^c=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | #27279935, #27279937, #27280993, … |
| maximal normal subgroup of | #27279935, #27279937, #27280993, … |
| maximal quotient of | #27384151, #27387140, #27388985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 14641);