Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #1201 |
| automorphism | Aut(G) = C24×D3×AGL(1,𝔽5) (order 1920) · Inn(G) = D3×D5 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C5 ⋊ C22⋊C4 |
| Sylow | C5 ⋊ (C2×D3)⋊C4 |
| Sylow | C3 ⋊ #685 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×D5 |
| commutator | C2×C3×C5 | C2×C4 |
| Fitting, socle, cosocle | C22×C3×C5 | C22 |
| prime radical | C22×D15 | C2 |
| hypocenter | C3×C5 | C22⋊C4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,5,3)(7,8)(9,11,12,10)(14,16);(2,3)(4,5)(6,7,8)(9,10,12,11)(13,14,15,16)Polycyclic presentation:
2^4,3,5|a^2=c,b^2=cd,b^a=bd,f^a=f^4,e^b=e^242 conjugacy classes — view the full character table.
| maximal subgroup of | #91326, #91337, #91345, … |
| maximal normal subgroup of | #91326, #91337, #91345, … |
| maximal quotient of | #90957, #90960, #90963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 28);