Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#390 ⋊ C2 |
| Sylow | C53 ⋊ #2554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C53×#390 | C2 |
| hypercenter | #390 | D53 |
| hypocenter | C53 | #2554 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #390 | |
560 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,b^2=eg,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^52,h^d=h^52766 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2554, C2×#20279772, #27554265, … |
| maximal normal subgroup of | D53×#2554, C2×#20279772, #27554265, … |
| maximal quotient of | #27306393, #27306552, #27307333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 9567);