Finite Group DatabasebetaFinite Group Databasebeta| order | 656 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊#4223 |
| unambiguous semidirect product | (C2×#4223)⋊C2 |
| Fitting | C22×C41 ⋊ C4 |
| Sylow | C41 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#4223 |
| commutator, socle | C2×C41 | C2×C4 |
| Fitting | C22×C41 | C4 |
| prime radical | C22×D41 | C2 |
| cosocle | C2×D41 | C22 |
| hypercenter | C22 | #4223 |
| hypocenter | C41 | C22⋊C4 |
| LCS | G ▷ C2×C41 ▷ C41 | |
| UCS | C1 ◁ C2 ◁ C22 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 164 |
Polycyclic presentation:
2^4,41|a^2=c,b^a=bd,e^a=e^32,e^c=e^4050 conjugacy classes — view the full character table.
| maximal subgroup of | #13050222, #13050223, C2×C22⋊#4223, … |
| maximal normal subgroup of | #13050222, #13050223, C2×C22⋊#4223, … |
| maximal quotient of | (C2×#4223)⋊C4, C22⋊#66493, #13050221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(656, 35);