Finite Group DatabasebetaFinite Group Databasebeta| order | 7856 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic491⋊C4 |
| Sylow | C491 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D491 |
| commutator | C2×C491 | C2×C4 |
| Fitting | C2×C4×C491 | C2 |
| socle | C22×C491 | C22 |
| hypercenter | C2×C4 | D491 |
| hypocenter | C491 | C4⋊C4 |
| LCS | G ▷ C2×C491 ▷ C491 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
26 conjugacy classes of subgroups.
Polycyclic presentation:
2^4,491|a^2=d,b^2=c,b^a=bc,e^b=e^4901970 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4⋊C4×D491, C2×Dic491⋊C4, #30628293, … |
| maximal normal subgroup of | C4⋊C4×D491, C2×Dic491⋊C4, #30628293, … |
| maximal quotient of | C491⋊#95, Dic491⋊C8, #30628268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7856, 19);