Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1379 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊D53 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C22×C8 | C2×D53 |
| Fitting | C53×#405 | C2 |
| socle | C22×C53 | C22⋊C8 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #1379 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ #405 | |
136 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 81 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,c^2=g,d^2=f,f^2=g,b^a=bc,h^a=h^52,c^b=cg,d^b=dg1096 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#1379, C2×#20272804, #27298963, … |
| maximal normal subgroup of | D53×#1379, C2×#20272804, #27298963, … |
| maximal quotient of | #27270931, #27270938, #27270945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2599);