Finite Group DatabasebetaFinite Group Databasebeta| order | 2142 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C17 · Schur multiplier (H2) = C1 |
| direct product | C2×C17×C7⋊C9 |
| unambiguous semidirect product | (C2×C7)⋊(C9×C17) |
| unambiguous semidirect product | (C2×C7×C17)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C9×C17 |
| Hall | C2×C7×C17 ⋊ C9 |
| Hall | C7×C17 ⋊ C2×C9 |
| Hall | C2×C7 ⋊ C9×C17 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C17 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C2×C9×C17 |
| Frattini | C3 | C2×C17×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C2×C3×C7×C17 | C3 |
| cosocle | C3×C7 | C2×C3×C17 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C17 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 2142 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22,23)(24,26,25)(27,28,30,29,31,33,32,34,35);(20,21,22,24,23,25,26)510 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7⋊C9×Dic17, C17×Dic7⋊C9, (C7⋊C9)⋊Dic17, … |
| maximal normal subgroup of | C7⋊C9×Dic17, C17×Dic7⋊C9, (C7⋊C9)⋊Dic17, … |
| maximal quotient of | C4×C17×C7⋊C9, C22×C17×C7⋊C9, C2×C17×C49⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2142, 5);