Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #1445 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D53 |
| commutator | C23×C53 | C22×C4 |
| Fitting | C22×C4⋊C4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C4⋊C4 | D53 |
| hypocenter | C53 | #1445 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C4⋊C4 | |
520 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 197 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=g,b^2=de,c^2=de,b^a=bd,c^a=ce,h^a=h^52,c^b=cf1084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27271414, #27276224, #27276226, … |
| maximal normal subgroup of | #27271414, #27276224, #27276226, … |
| maximal quotient of | #27272016, #27272019, #27292822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2778);