Finite Group DatabasebetaFinite Group Databasebeta| order | 456 |
| isoclinism | isoclinic class of D19⋊C3 |
| automorphism | Aut(G) = C22×AGL(1,𝔽19) (order 1368) · Inn(G) = D19⋊C3 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×D19⋊C3 |
| unambiguous semidirect product | D19⋊(C3×C4) |
| unambiguous semidirect product | (C4×D19)⋊C3 |
| Fitting | C4×C19 ⋊ C2×C3 |
| derived, Sylow | C19 ⋊ C2×C3×C4 |
| Hall | C4×D19 ⋊ C3 |
| Hall | C19⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D19⋊C3 |
| commutator, hypocenter | C19 | C2×C3×C4 |
| Frattini | C2 | C2×D19⋊C3 |
| Fitting | C4×C19 | C2×C3 |
| socle, cosocle | C2×C19 | C22×C3 |
| prime radical | C2×D19⋊C3 | C2 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 76 |
A minimal faithful permutation representation — 2 generators on 23 points:
(6,10,11,7,8,12)(9,18,15,13,14,20)(16,21,23,22,19,17);(1,2,3,4)(5,6,9,15,21,18,22,8,12,17,23,11,10,16,14,19,20,13,7)36 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D19⋊C3, #64515⋊C3, C2×C4×D19⋊C3, … |
| maximal normal subgroup of | C8×D19⋊C3, #64515⋊C3, C2×C4×D19⋊C3, … |
| maximal quotient of | C8×D19⋊C3, #64515⋊C3, C4×(C19⋊C3)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(456, 8);