Finite Group DatabasebetaFinite Group Databasebeta| order | 816 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D51⋊Q8 |
| Hall | C3×C17 ⋊ C2×Q8 |
| Sylow | C17 ⋊ D3×Q8 |
| Sylow | C3 ⋊ Q8×D17 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D17 |
| commutator, socle, cosocle | C2×C3×C17 | C23 |
| Fitting | C3×C4×C17 | C22 |
| prime radical | C2×D51 | C22 |
| hypercenter | C4 | D3×D17 |
| hypocenter | C3×C17 | C2×Q8 |
| LCS | G ▷ C2×C3×C17 ▷ C3×C17 | |
| UCS | C1 ◁ C2 ◁ C4 | |
76 conjugacy classes of subgroups.
| transitive degree | μt(G) = 408 |
Polycyclic presentation:
2^4,3,17|a^2=d,b^2=d,c^2=d,c^a=cd,f^a=f^16,c^b=cd,e^b=e^287 conjugacy classes — view the full character table.
| maximal subgroup of | #13108992, #13109001, #13109014, … |
| maximal normal subgroup of | #13108992, #13109001, #13109014, … |
| maximal quotient of | #13109019, #13109023, #13109026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(816, 127);