Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#364 ⋊ C2 |
| Sylow | C53 ⋊ #2170 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | D4×D53 |
| commutator | C4×C53 | C22×C8 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C53×#364 | C2 |
| socle | C2×C53 | C8×D4 |
| hypercenter | #364 | D53 |
| hypocenter | C53 | #2170 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ #364 | |
146 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 63 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,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,d^b=dg,h^b=h^52,h^c=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | #27295845, #27295847, #27295850, … |
| maximal normal subgroup of | #27295845, #27295847, #27295850, … |
| maximal quotient of | #27295796, #27295808, #27295857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6346);