Finite Group DatabasebetaFinite Group Databasebeta| order | 7552 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C59×#340 ⋊ C2 |
| Sylow | C59 ⋊ #2054 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | D4×D59 |
| commutator | C22×C4×C59 | C23 |
| Frattini | C22×C4 | C22×D59 |
| Fitting | C59×#340 | C2 |
| socle | C23×C59 | C2×D4 |
| hypercenter | #340 | D59 |
| hypocenter | C59 | #2054 |
| LCS | G ▷ C22×C4×C59 ▷ C2×C59 ▷ C59 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ #340 | |
360 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,59|c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^58,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=h^58,h^c=h^58844 conjugacy classes — view the full character table.
| maximal subgroup of | #28501131, #28501136, #28501142, … |
| maximal normal subgroup of | #28501131, #28501136, #28501142, … |
| maximal quotient of | #28492482, #28492489, #28492566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7552, 5822);