Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #67194 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Hall | C32×C5 ⋊ C16 |
| Sylow | C32 ⋊ #672 |
| Sylow | C5 ⋊ #3771 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #67194 |
| commutator, hypocenter | C32×C5 | C16 |
| Fitting, socle, prime radical | C2×C32×C5 | C8 |
| cosocle | C5×#622 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
26 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^4,3^2,5|a^2=b,b^2=c,c^2=d,e^a=e^2f^2,f^a=ef^2,g^a=g^4,e^b=f^2,f^b=e,e^c=e^2,f^c=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3771, #13088387, #13088388, … |
| maximal normal subgroup of | D5×#3771, #13088387, #13088388, … |
| maximal quotient of | #13085378, C2×#10623954 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 423);