Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | #124×C53 ⋊ C4 |
| Sylow | C53 ⋊ #2337 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #124×C53 | C4 |
| hypercenter | #124 | #6157 |
| hypocenter | C53 | #2337 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
540 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=eg,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,h^a=h^23,c^b=ce,h^b=h^30,h^g=h^52226 conjugacy classes — view the full character table.
| maximal subgroup of | #28305052, #28305053, #28305056, … |
| maximal normal subgroup of | #28305052, #28305053, #28305056, … |
| maximal quotient of | #28312749, #28312764, #28312768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21011);