Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #1313 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×#6157 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C2×C8 | C2×#6157 |
| Fitting | C2×C16×C53 | C4 |
| socle | C22×C53 | #97 |
| hypercenter | C2×C16 | #6157 |
| hypocenter | C53 | #1313 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C16 | |
80 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,b^a=be,d^a=dg,c^b=cg,h^b=h^23,h^d=h^52460 conjugacy classes — view the full character table.
| maximal subgroup of | #28298870, #28298874, C2×#20289632, … |
| maximal normal subgroup of | #28298870, #28298874, C2×#20289632, … |
| maximal quotient of | #28298089, #28298091, #28298257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19427);