Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2971 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #117×C53 | C4 |
| hypercenter | #117 | #6157 |
| hypocenter | C53 | #2971 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 | |
402 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 160 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=f,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cg,d^a=dg,h^a=h^30,c^b=cg,d^c=dg,h^c=h^30,h^d=h^52,h^f=h^52310 conjugacy classes — view the full character table.
| maximal subgroup of | #28304984, #28304986, #28304992, … |
| maximal normal subgroup of | #28304984, #28304986, #28304992, … |
| maximal quotient of | #28318491, #28318493, #28318564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21353);