Finite Group DatabasebetaFinite Group Databasebeta| order | 408 |
| isoclinism | isoclinic class of D3×#595 |
| automorphism | Aut(G) = D3×AGL(1,𝔽17) (order 1632) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | D3×#595 |
| unambiguous semidirect product | (C3×#595)⋊C2 |
| unambiguous semidirect product | (C3⋊#595)⋊C2 |
| Fitting, socle, derived, Hall | C3×C17 ⋊ C2×C4 |
| Sylow | C17 ⋊ C4×D3 |
| Sylow | C3 ⋊ C2×#595 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C17 | C2×C4 |
| prime radical | D3×D17 | C2 |
| cosocle | C3×D17 | C22 |
| LCS | G ▷ C3×C17 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 51 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(18,19,20);(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)(19,20)24 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3633, C2×D3×#595, D9×#595, … |
| maximal normal subgroup of | D3×#3633, C2×D3×#595, C3×D3×#595 |
| maximal quotient of | Dic3×#595, (C2×D3)⋊#595, Dic3⋊#595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(408, 35);