Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #1308 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133124 |
| commutator | C23×C53 | C42 |
| Frattini | C2×C42 | C2×D53 |
| Fitting | C53×#432 | C2 |
| socle | C22×C53 | #95 |
| hypercenter | #432 | D53 |
| hypocenter | C53 | #1308 |
| LCS | G ▷ C23×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #432 | |
122 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 53 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=fg,e^2=f,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df,h^b=h^52754 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1308, C2×#20272627, #27274494, … |
| maximal normal subgroup of | D53×#1308, C2×#20272627, #27274494, … |
| maximal quotient of | #27270664, #27270665, #27270666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2422);