Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C53 ⋊ C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#6157 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | #95×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | #95 | #6157 |
| hypocenter | C53 | C8×C22⋊C4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #95 | |
384 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 134 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,b^2=f,c^2=e,e^2=g,b^a=bd,h^a=h^23,h^c=h^30,h^e=h^52340 conjugacy classes — view the full character table.
| maximal subgroup of | #28301908, #28301911, #28301920, … |
| maximal normal subgroup of | #28301908, #28301911, #28301920, … |
| maximal quotient of | #28301491, #28301500, #28301514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20244);