Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×#123×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2534 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C2×#123×C53 | C2 |
| hypercenter | C2×#123 | D53 |
| hypocenter | C53 | #2534 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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=eg,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^52,h^c=h^52766 conjugacy classes — view the full character table.
| maximal subgroup of | #27302381, #27302394, #27437481, … |
| maximal normal subgroup of | #27302381, #27302394, #27437481, … |
| maximal quotient of | #27306316, #27306810, #27306867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 9399);