Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | #339×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1764 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4×D67 |
| commutator | C4×C67 | C2×C42 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #339×C67 | C2 |
| socle | C22×C67 | C4×D4 |
| hypercenter | #339 | D67 |
| hypocenter | C67 | #1764 |
| LCS | G ▷ C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ #339 | |
300 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|a^2=e,b^2=f,c^2=e,d^2=g,e^2=g,b^a=bd,d^a=dg,f^a=fg,h^c=h^66980 conjugacy classes — view the full character table.
| maximal subgroup of | #1764×D67, C2×#22560043, #30853674, … |
| maximal normal subgroup of | #1764×D67, C2×#22560043, #30853674, … |
| maximal quotient of | #30746502, #30746512, #30746519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4344);