Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of (C2×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×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C5 ⋊ M16 |
| Sylow | C5 ⋊ ΓL(1,𝔽25) |
| Sylow | C3 ⋊ #700 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AGL(1,𝔽5) |
| commutator, Fitting, socle | C2×C3×C5 | C2×C4 |
| prime radical | C2×D15 | C4 |
| cosocle | C3×Dic5 | C22 |
| hypocenter | C3×C5 | M16 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 | |
40 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)(10,14,15,16);(2,3,4,5)(6,7,8)(9,10,12,14,13,15,11,16)Polycyclic presentation:
2^4,3,5|a^2=c,b^2=d,c^2=d,b^a=bd,f^a=f^2,e^b=e^2,f^c=f^424 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#700, #91914, #91916, … |
| maximal normal subgroup of | D3×#700, #91914, #91916, … |
| maximal quotient of | #91169, #91171, Dic3⋊#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 101);