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×Dic53 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C4×C8 | C2×D53 |
| Fitting | C53×#346 | C2 |
| socle | C22×C53 | #97 |
| hypercenter | #346 | D53 |
| hypocenter | C53 | #1313 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #346 | |
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^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27271427, #27271429, D53×#1313, … |
| maximal normal subgroup of | #27271427, #27271429, D53×#1313, … |
| maximal quotient of | #27270708, #27270921, #27271427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2430);