Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#364 ⋊ C2 |
| Sylow | C53 ⋊ #2153 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | D4×D53 |
| commutator | C4×C53 | C22×C8 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C53×#364 | C2 |
| socle | C2×C53 | C2×C4⋊C8 |
| hypercenter | #364 | D53 |
| hypocenter | C53 | #2153 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ #364 | |
152 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 75 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=e,b^2=d,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,c^b=cg,h^c=h^52772 conjugacy classes — view the full character table.
| maximal subgroup of | #27295606, #27295608, #27295611, … |
| maximal normal subgroup of | #27295606, #27295608, #27295611, … |
| maximal quotient of | #27295638, #27295664, #27295697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6274);