Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of #4177⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4177⋊D5 |
| Fitting, derived, Hall | C5×#731 ⋊ C22 |
| Sylow | #731 ⋊ C2×D5 |
| Sylow | C5 ⋊ #4176⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C5×#731 | C22 |
| Frattini | C32 | D3×D15 |
| socle | C3×C5 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ C5×#731 | |
| UCS | C1 | |
108 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
Polycyclic presentation:
2^2,3^2,5,3^2|d^3=g^2,c^a=c^2,e^a=e^4,f^a=f^2,d^b=d^2g,f^b=f^2,g^b=g^2,d^c=dfg,f^c=fg^243 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#4176⋊C2, #72268692 |
| maximal normal subgroup of | D5×#4176⋊C2, #72268692 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 138);