Finite Group DatabasebetaFinite Group Databasebeta| order | 408 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×C16 (order 384) · Inn(G) = D3 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C8×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C8×C17) |
| derived, Sylow | C3 ⋊ C8×C17 |
| Hall | C3×C17 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C17 | D3 |
| commutator, hypocenter | C3 | C8×C17 |
| Frattini | C4 | D3×C17 |
| Fitting | C3×C4×C17 | C2 |
| socle, prime radical | C2×C3×C17 | C4 |
| cosocle | C3×C4 | C2×C17 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C17 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 408 |
A minimal faithful permutation representation — 2 generators on 28 points:
(2,3)(4,5,6,8,7,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28);(1,2,3)204 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×D17, #11717739, #11717740, … |
| maximal normal subgroup of | C3⋊C8×D17, #11717739, #11717740, … |
| maximal quotient of | C17×C3⋊C16, C2×C17×C3⋊C8, C17×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(408, 1);