Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×M16×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1728 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×#6157 |
| commutator, socle | C22×C53 | C2×C42 |
| Fitting | C2×M16×C53 | C4 |
| hypercenter | C2×M16 | #6157 |
| hypocenter | C53 | #1728 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M16 | |
480 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 146 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,e^2=g,b^a=bd,c^a=cg,h^b=h^30,h^f=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | #28298429, #28298435, #28306537, … |
| maximal normal subgroup of | #28298429, #28298435, #28306537, … |
| maximal quotient of | #28300063, #28300064, #28300067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20172);