Finite Group DatabasebetaFinite Group Databasebeta| order | 204 |
| isoclinism | isoclinic class of D3×D17 |
| automorphism | Aut(G) = D3×AGL(1,𝔽17) (order 1632) · Inn(G) = G · Out(G) = C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D3×D17 |
| unambiguous semidirect product | (D3×C17)⋊C2 |
| unambiguous semidirect product | (C3×D17)⋊C2 |
| unambiguous semidirect product | D51⋊C2 |
| Fitting, socle, derived, Hall | C3×C17 ⋊ C22 |
| Sylow | C17 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D17 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C17 | C22 |
| LCS | G ▷ C3×C17 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 51 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19,20);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(19,20)30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#595, C2×D3×D17, D9×D17, … |
| maximal normal subgroup of | D3×#595, C2×D3×D17, C3×D3×D17, … |
| maximal quotient of | Dic3×D17, D3×Dic17, #87972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(204, 7);