Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽5) (order 480) · Inn(G) = D3×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Hall | C3×C5 ⋊ C2×C8 |
| Sylow | C5 ⋊ D3×C8 |
| Sylow | C3 ⋊ D5⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AGL(1,𝔽5) |
| commutator, hypocenter | C3×C5 | C2×C8 |
| Fitting, socle | C2×C3×C5 | C2×C4 |
| prime radical | C2×D15 | C4 |
| cosocle | C3×Dic5 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,5,4)(7,8)(9,11,13,12)(10,14,16,15);(2,3,4,5)(6,7,8)(9,10,12,15,13,16,11,14)Polycyclic presentation:
2^4,3,5|a^2=c,b^2=d,c^2=d,f^a=f^2,e^b=e^2,f^c=f^430 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5⋊C8, #91914, #91915, … |
| maximal normal subgroup of | D3×D5⋊C8, #91914, #91915, … |
| maximal quotient of | #91164, #91166, Dic3×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 99);