Finite Group DatabasebetaFinite Group Databasebeta| order | 456 |
| isoclinism | isoclinic class of C19⋊C3 |
| automorphism | Aut(G) = C22×AGL(1,𝔽19) (order 1368) · Inn(G) = C19⋊C3 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C8×C19⋊C3 |
| unambiguous semidirect product | (C8×C19)⋊C3 |
| Fitting, Hall | C8×C19 ⋊ C3 |
| derived, Sylow | C19 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | C19⋊C3 |
| commutator, hypocenter | C19 | C3×C8 |
| Frattini | C4 | C2×C19⋊C3 |
| Fitting | C8×C19 | C3 |
| socle | C2×C19 | C3×C4 |
| prime radical | C2×C19⋊C3 | C4 |
| cosocle | C4×C19 | C2×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C8 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 152 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,5,4,6,7,8)(10,11,13)(12,15,16)(14,18,17)(19,23,21)(20,24,22)(25,27,26);(9,10,12,16,21,26,23,11,14,19,24,13,17,22,15,20,25,27,18)72 conjugacy classes — view the full character table.
| maximal subgroup of | (C19⋊C3)⋊C16, C16×C19⋊C3, C8×D19⋊C3, … |
| maximal normal subgroup of | (C19⋊C3)⋊C16, C16×C19⋊C3, C8×D19⋊C3, … |
| maximal quotient of | C16×C19⋊C3, C2×C8×C19⋊C3, C8×#4294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(456, 2);