Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×#126×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2603 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C2×#126×C53 | C2 |
| hypercenter | C2×#126 | D53 |
| hypocenter | C53 | #2603 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
582 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|b^2=e,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=de,h^a=h^52,c^b=cg,h^c=h^52766 conjugacy classes — view the full character table.
| maximal subgroup of | #27357870, #27357874, #27357900, … |
| maximal normal subgroup of | #27357870, #27357874, #27357900, … |
| maximal quotient of | #27306756, #27306805, #27308230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 10139);