Finite Group DatabasebetaFinite Group Databasebeta| order | 2070 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C23 · Schur multiplier (H2) = C3 |
| direct product | C5×C3⋊D3×C23 |
| unambiguous semidirect product | C3⋊(C5×D3×C23) |
| Fitting, socle, Hall | C32×C5×C23 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C5×C23 |
| Hall | C32×C23 ⋊ C2×C5 |
| Hall | C32×C5 ⋊ C2×C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C23 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C2×C5×C23 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C23 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C5×C23 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 1035 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2,3,4,5)(7,8)(10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(9,10,11);(6,7,8)690 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×C3⋊D3×D23, D5×C3⋊D3×C23, C3⋊D3×D115, … |
| maximal normal subgroup of | C5×C3⋊D3×D23, D5×C3⋊D3×C23, C3⋊D3×D115, … |
| maximal quotient of | C5×C23×C3⋊Dic3, C2×C5×C3⋊D3×C23, C5×C23×C3⋊D21, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2070, 17);