Finite Group DatabasebetaFinite Group Databasebeta| order | 408 |
| isoclinism | isoclinic class of #595 |
| automorphism | Aut(G) = C22×AGL(1,𝔽17) (order 1088) · Inn(G) = #595 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×#595 |
| unambiguous semidirect product | D17⋊(C3×C4) |
| Fitting, socle | C2×C3×C17 ⋊ C4 |
| derived, Sylow | C17 ⋊ C2×C3×C4 |
| Hall | C3×C17 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #595 |
| commutator, hypocenter | C17 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C17 | C4 |
| prime radical | C2×C3×D17 | C2 |
| cosocle | D17 | C22×C3 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C2×C3 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 102 |
A minimal faithful permutation representation — 2 generators on 22 points:
(3,4,5)(7,8,9,11)(10,13,14,18)(12,17,15,16)(19,21,22,20);(1,2)(6,7,10,15,11,16,20,22,18,13,19,21,17,8,12,14,9)48 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#595, (C2×D3)⋊#595, Dic3⋊#595, … |
| maximal normal subgroup of | Dic3×#595, (C2×D3)⋊#595, Dic3⋊#595, … |
| maximal quotient of | C3×#63242, C3×#63243, C3×C4×#595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(408, 39);