Finite Group DatabasebetaFinite Group Databasebeta| order | 1596 |
| isoclinism | isoclinic class of D57 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×Dic57 |
| unambiguous semidirect product | C3⋊(C7×Dic19) |
| unambiguous semidirect product | C19⋊(C7×Dic3) |
| derived, Hall | C3×C19 ⋊ C4×C7 |
| Hall | C3×C7×C19 ⋊ C4 |
| Hall | C7×C19 ⋊ Dic3 |
| Hall | C3×C7 ⋊ Dic19 |
| Sylow | C19 ⋊ C7×Dic3 |
| Sylow | C3 ⋊ C7×Dic19 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D57 |
| commutator, hypocenter | C3×C19 | C4×C7 |
| Frattini | C2 | C7×D57 |
| Fitting, socle, prime radical | C2×C3×C7×C19 | C2 |
| cosocle | C2×C3×C19 | C2×C7 |
| LCS | G ▷ C3×C19 | |
| UCS | C1 ◁ C2×C7 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1596 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33);(8,10,9,11)(13,14)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)420 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×Dic171, C7×C3⋊((C19⋊C3)⋊C4), C3⋊#13097582, … |
| maximal normal subgroup of | C7×C3⋊((C19⋊C3)⋊C4), C3⋊#13097582, C3⋊#13097583, … |
| maximal quotient of | C7×Dic171, C3×C7×Dic57, C7×C3⋊Dic57, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1596, 21);