Finite Group DatabasebetaFinite Group Databasebeta| order | 456 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×D3×C9 (order 432) · Inn(G) = D3 · Out(G) = C23×C9 |
| homology | Abelianization (H1) = C8×C19 · Schur multiplier (H2) = C1 |
| direct product | C19×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C8×C19) |
| derived, Sylow | C3 ⋊ C8×C19 |
| Hall | C3×C19 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C19 | D3 |
| commutator, hypocenter | C3 | C8×C19 |
| Frattini | C4 | D3×C19 |
| Fitting | C3×C4×C19 | C2 |
| socle, prime radical | C2×C3×C19 | C4 |
| cosocle | C3×C4 | C2×C19 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C19 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 456 |
A minimal faithful permutation representation — 2 generators on 30 points:
(2,3)(4,8,10,5,7,11,6,9)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30);(1,2,3)228 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×D19, D57⋊C8, #11744815, … |
| maximal normal subgroup of | C3⋊C8×D19, D57⋊C8, #11744815, … |
| maximal quotient of | C19×C3⋊C16, C2×C19×C3⋊C8, C19×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(456, 3);