Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#405 ⋊ C2 |
| Sylow | C53 ⋊ #2951 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#405 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #2951 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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,c^2=f,d^2=g,e^2=g,b^a=be,d^a=dg,e^a=eg,h^a=h^52,e^b=eg,h^b=h^52,d^c=dg1084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27300753, #27300757, #27300760, … |
| maximal normal subgroup of | #27300753, #27300757, #27300760, … |
| maximal quotient of | #27353575, #27353803, #27353946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13387);