Finite Group DatabasebetaFinite Group Databasebeta| order | 1596 |
| isoclinism | isoclinic class of D19⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×(C19⋊C3)⋊C4 |
| unambiguous semidirect product | (C19⋊C3)⋊(C4×C7) |
| unambiguous semidirect product | Dic19⋊(C3×C7) |
| derived, Sylow | C19 ⋊ C3×C4×C7 |
| Hall | C7×Dic19 ⋊ C3 |
| Hall | C7×C19⋊C3 ⋊ C4 |
| Hall | C7×C19 ⋊ C3×C4 |
| Hall | Dic19 ⋊ C3×C7 |
| Hall | C19⋊C3 ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D19⋊C3 |
| commutator, hypocenter | C19 | C3×C4×C7 |
| Frattini | C2 | C7×D19⋊C3 |
| Fitting, socle | C2×C7×C19 | C2×C3 |
| prime radical | C2×C7×C19⋊C3 | C2 |
| cosocle | C2×C19 | C2×C3×C7 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2×C7 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 532 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5,6,7)(9,13,14,10,11,15)(12,21,18,16,17,23)(19,24,26,25,22,20)(27,28,29,30);(8,9,12,18,24,21,25,11,15,20,26,14,13,19,17,22,23,16,10)126 conjugacy classes — view the full character table.
| maximal subgroup of | C7×#4296⋊C4, #18012157, #18012158, … |
| maximal normal subgroup of | C7×#4296⋊C4, #18012157, #18012158, … |
| maximal quotient of | C7×#4294⋊C4, C3×C7×(C19⋊C3)⋊C4, C7×C3⋊((C19⋊C3)⋊C4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1596, 1);