Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1378 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #11719921 |
| commutator | C22×C53 | C4×C8 |
| Frattini | C22×C8 | C2×D53 |
| Fitting | C2×M32×C53 | C2 |
| socle | C2×C53 | #336 |
| hypercenter | C2×M32 | D53 |
| hypocenter | C53 | #1378 |
| LCS | G ▷ C22×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×M32 | |
132 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 73 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=d,b^2=e,d^2=f,f^2=g,b^a=bc,e^a=eg,h^a=h^52,c^b=cg,h^b=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#1378, C2×#20272802, #27299116, … |
| maximal normal subgroup of | D53×#1378, C2×#20272802, #27299116, … |
| maximal quotient of | #27270929, #27270939, #27270946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2597);