Finite Group DatabasebetaFinite Group Databasebeta| order | 816 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C17⋊(#41⋊C3) |
| unambiguous semidirect product | #63256⋊C3 |
| Hall | #63256 ⋊ C3 |
| Sylow | C17 ⋊ #41⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D17 |
| commutator, Fitting, cosocle, hypocenter | Q8×C17 | C2×C3 |
| socle | C2×C17 | C2×A4 |
| LCS | G ▷ Q8×C17 | |
| UCS | C1 ◁ C2 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 272 |
Polycyclic presentation:
2,3,2^2,17,2|a^2=f,c^2=f,d^2=f,e^a=e^16,c^b=cd,d^b=c,d^c=df70 conjugacy classes — view the full character table.
| maximal subgroup of | #13109613, #10587680⋊C3, #13109619, … |
| maximal normal subgroup of | #13109613, #10587680⋊C3, #13109619, … |
| maximal quotient of | SL(2,𝔽3)×Dic17, C2×C17⋊(#41⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(816, 104);