Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C53 ⋊ #1309 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133124 |
| commutator | C23×C53 | C42 |
| Frattini | C2×C42 | C2×D53 |
| Fitting | C53×#423 | C2 |
| socle | C22×C53 | #95 |
| hypercenter | #423 | D53 |
| hypocenter | C53 | #1309 |
| LCS | G ▷ C23×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #423 | |
146 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df,h^b=h^52754 conjugacy classes — view the full character table.
| maximal subgroup of | #27271635, #27271636, #27271713, … |
| maximal normal subgroup of | #27271635, #27271636, #27271713, … |
| maximal quotient of | #27270667, #27270668, #27270688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2423);