Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C53×#396 ⋊ C2 |
| Sylow | C53 ⋊ #2576 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#396 | C2 |
| hypercenter | #396 | D53 |
| hypocenter | C53 | #2576 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #396 | |
468 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=eg,b^2=efg,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg,h^d=h^52610 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2576, C2×#20280012, #27554575, … |
| maximal normal subgroup of | D53×#2576, C2×#20280012, #27554575, … |
| maximal quotient of | #27306430, #27306699, #27307056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 9807);