Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#431 ⋊ C2 |
| Sylow | C53 ⋊ #1476 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊D53 |
| commutator | C4×C53 | C22×C8 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#431 | C2 |
| socle | C22×C53 | C2×C22⋊C4 |
| hypercenter | #431 | D53 |
| hypocenter | C53 | #1476 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ #431 | |
276 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 101 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=f,d^2=f,e^2=g,b^a=bd,d^a=df,d^b=df,h^b=h^521096 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27274763, #27274767, #27274768, … |
| maximal normal subgroup of | #27274763, #27274767, #27274768, … |
| maximal quotient of | #27272029, #27272037, #27272042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2890);