Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#413 ⋊ C2 |
| Sylow | C53 ⋊ #3018 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133187 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#413 | C2 |
| socle | C2×C53 | C2×#120 |
| hypercenter | #413 | D53 |
| hypocenter | C53 | #3018 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #413 | |
724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^52,e^b=eg,f^b=fg,h^b=h^52,e^c=eg604 conjugacy classes — view the full character table.
| maximal subgroup of | #27301413, #27301418, #27301423, … |
| maximal normal subgroup of | #27301413, #27301418, #27301423, … |
| maximal quotient of | #27356202, #27356253, #27356304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13863);