Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C5⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ C4⋊C4 |
| Sylow | C32 ⋊ #683 |
| Sylow | C5 ⋊ #3777 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊PSU(3,𝔽2) |
| commutator | C5×C3⋊D3 | C2×C4 |
| Fitting, socle | C2×C32×C5 | Q8 |
| prime radical, cosocle | C2×C5×C3⋊D3 | C22 |
| hypocenter | C32×C5 | C4⋊C4 |
| LCS | G ▷ C5×C3⋊D3 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
62 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,b^2=cd,b^a=bc,e^a=f,f^a=e^2,g^a=g^4,e^b=ef^2,f^b=e^2f^2,e^c=e^2,f^c=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3777, #13088482, #13088483, … |
| maximal normal subgroup of | D5×#3777, #13088482, #13088483, … |
| maximal quotient of | #13085624, #13085626, #13085628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 485);