Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C8×Q8×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2982 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator | C2×C53 | C23×C8 |
| Fitting | C8×Q8×C53 | C2 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C8×Q8 | D53 |
| hypocenter | C53 | #2982 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C8×Q8 | |
396 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 217 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=e,f^2=g,b^a=be,d^a=de,c^b=ce,h^d=h^521108 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27353534, #27353536, #27353538, … |
| maximal normal subgroup of | #27353534, #27353536, #27353538, … |
| maximal quotient of | #27382820, #27383130, #27383131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13589);