Finite Group DatabasebetaFinite Group Databasebeta| order | 456 |
| isoclinism | isoclinic class of C22⋊(C19⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C22⋊(C19⋊C3) |
| unambiguous semidirect product | C23⋊(C19⋊C3) |
| unambiguous semidirect product | (C2×C19)⋊A4 |
| Fitting, socle, Hall | C23×C19 ⋊ C3 |
| derived | C22×C19 ⋊ C2×C3 |
| Sylow | C19 ⋊ C2×A4 |
| Sylow | C23 ⋊ C19⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C19⋊C3) |
| commutator, cosocle, hypocenter | C22×C19 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C19 | C3 |
| LCS | G ▷ C22×C19 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 114 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2)(4,5,6)(8,9,11)(10,13,14)(12,16,15)(17,21,19)(18,22,20)(23,25,24);(3,6)(4,5)(7,8,10,14,19,24,21,9,12,17,22,11,15,20,13,18,23,25,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | Dic19⋊A4, C4×C22⋊(C19⋊C3), C2×D19⋊A4, … |
| maximal normal subgroup of | Dic19⋊A4, C4×C22⋊(C19⋊C3), C2×D19⋊A4, … |
| maximal quotient of | C4×C22⋊(C19⋊C3), C2×Q8⋊(C19⋊C3), #41⋊(C19⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(456, 50);