Finite Group DatabasebetaFinite Group Databasebeta| order | 2070 |
| isoclinism | isoclinic class of C3⋊D345 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊D345 |
| unambiguous semidirect product | C5⋊(C3⋊D69) |
| unambiguous semidirect product | C23⋊(C3⋊D15) |
| Fitting, socle, derived, Hall | C32×C5×C23 ⋊ C2 |
| Hall | C32×C23 ⋊ D5 |
| Hall | C5×C23 ⋊ C3⋊D3 |
| Hall | C32×C5 ⋊ D23 |
| Sylow | C23 ⋊ C3⋊D15 |
| Sylow | C32 ⋊ D115 |
| Sylow | C5 ⋊ C3⋊D69 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C5×C23 | C2 |
| LCS | G ▷ C32×C5×C23 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1035 |
Polycyclic presentation:
2,3^2,5,23|b^a=b^2,c^a=c^2,d^a=d^4,e^a=e^22519 conjugacy classes — view the full character table.
| maximal subgroup of | D23×C3⋊D15, D5×C3⋊D69, C3⋊D3×D115, … |
| maximal normal subgroup of | D23×C3⋊D15, D5×C3⋊D69, C3⋊D3×D115, … |
| maximal quotient of | C3⋊Dic345, C2×C3⋊D345, C3⋊D2415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2070, 20);