Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×M32×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2169 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D53 |
| commutator, socle | C22×C53 | C22×C8 |
| Fitting | C2×M32×C53 | C2 |
| hypercenter | C2×M32 | D53 |
| hypocenter | C53 | #2169 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C2×M32 | |
162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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=e,e^2=f,f^2=g,b^a=bd,c^a=cg,c^b=cg,h^b=h^521096 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#2169, C2×#20276545, #27537269, … |
| maximal normal subgroup of | D53×#2169, C2×#20276545, #27537269, … |
| maximal quotient of | #27295807, #27295877, #27295879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6340);