Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C32×C5 ⋊ #135 |
| Sylow | C32 ⋊ #4065 |
| Sylow | C5 ⋊ #63806 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10624033 |
| commutator, Fitting, cosocle | C32×C4×C5 | C23 |
| Frattini, hypercenter | C4 | D32×D5 |
| socle | C2×C32×C5 | C2×D4 |
| hypocenter | C32×C5 | #135 |
| LCS | G ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
268 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^5,3^2,5|a^2=d,b^2=d,c^2=e,d^2=e,c^a=cde,h^a=h^4,c^b=cd,f^b=f^2,d^c=de,g^c=g^293 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2070);