Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #6758 |
| automorphism | Aut(G) = D3×D4×AGL(1,𝔽5) (order 960) · Inn(G) = C2×ΓL(1,𝔽16) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊ΓL(1,𝔽16) |
| Fitting | C3×C4×C5 ⋊ C4 |
| Hall | C3×C5 ⋊ C4⋊C4 |
| Sylow | C5 ⋊ C4⋊Dic3 |
| Sylow | C3 ⋊ C4⋊AGL(1,𝔽5) |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,5,3,4)(6,7,8)(9,10,11,12);(2,3,4,5)(7,8)(10,12)Polycyclic presentation:
2^4,3,5|a^2=c,b^2=d,b^a=bd,e^a=e^2,f^a=f^3,f^c=f^430 conjugacy classes — view the full character table.
| maximal subgroup of | #91152, #91154, #91159, … |
| maximal normal subgroup of | #91152, #91154, #91159, … |
| maximal quotient of | #91222, #91223, #91224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 121);