Finite Group DatabasebetaFinite Group Databasebeta| order | 1980 |
| isoclinism | isoclinic class of D495 |
| automorphism | Aut(G) order 237600 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D990 |
| direct product | C2×D495 |
| unambiguous semidirect product | (C2×C5)⋊D99 |
| unambiguous semidirect product | (C2×C9)⋊D55 |
| unambiguous semidirect product | (C2×C11)⋊D45 |
| Fitting | C2×C5×C9×C11 ⋊ C2 |
| derived, Hall | C5×C9×C11 ⋊ C22 |
| Hall | C9×C11 ⋊ C2×D5 |
| Hall | C5×C11 ⋊ C2×D9 |
| Hall | C5×C9 ⋊ C2×D11 |
| Sylow | C11 ⋊ C2×D45 |
| Sylow | C9 ⋊ C2×D55 |
| Sylow | C5 ⋊ C2×D99 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D495 |
| commutator, cosocle, hypocenter | C5×C9×C11 | C22 |
| Frattini | C3 | C2×D165 |
| Fitting | C2×C5×C9×C11 | C2 |
| socle | C2×C3×C5×C11 | D3 |
| LCS | G ▷ C5×C9×C11 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 990 |
A minimal faithful permutation representation — 2 generators on 27 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);(4,11)(5,10)(6,9)(7,8)(13,16)(14,15)(18,27)(19,26)(20,25)(21,24)(22,23)498 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D7×D495, C2×C7×D495, C2×D3465, … |
| maximal normal subgroup of | C2×C7×D495, C2×C11×D495, C2×C13×D495, … |
| maximal quotient of | D7×D495, C2×C7×D495, C2×D3465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1980, 37);