Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2960 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C2×C4×C8×C53 | C2 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | #2960 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 | |
364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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,c^2=ef,f^2=g,b^a=be,d^a=dg,h^a=h^521720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27275101, #27275111, #27275124, … |
| maximal normal subgroup of | #27275101, #27275111, #27275124, … |
| maximal quotient of | #27382043, #27382059, #27382708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13444);