Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#376 ⋊ C2 |
| Sylow | C53 ⋊ #2401 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator, socle | C23×C53 | C24 |
| Frattini | C23 | C23×D53 |
| Fitting | C53×#376 | C2 |
| hypercenter | #376 | D53 |
| hypocenter | C53 | #2401 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
432 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 141 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=fg,b^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,h^c=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27392346, #27392349, #27392353, … |
| maximal normal subgroup of | #27392346, #27392349, #27392353, … |
| maximal quotient of | #27305787, #27305815, #27305871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7808);