Finite Group DatabasebetaFinite Group Databasebeta| order | 240 |
| isoclinism | isoclinic class of #6728 |
| automorphism | Aut(G) = C2×S5 (order 240) · Inn(G) = S5 · Out(G) = C2 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | S5 |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽5) | C2 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,47,7,48,3,35,19,36)(2,33,13,30,4,41,21,42)(5,34,20,29,15,44,10,39)(6,27,17,43,18,25,9,31)(8,40,14,46,16,32,22,38)(11,37,12,28,23,45,24,26);(1,29,3,39)(2,34,4,44)(5,26,15,28)(6,31,18,43)(7,36,19,48)(8,41,16,33)(9,46,17,38)(10,27,20,25)(11,32,23,40)(12,37,24,45)(13,42,21,30)(14,47,22,35)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 |
| order | 1 | 2 | 3 | 4 | 6 | 12 | 12 | 5 | 10 | 4 | 8 | 8 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 4 | 4 | 1 | 2 | 1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 |
| χ4 | 4 | 4 | 1 | -2 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 |
| χ5 | 4 | -4 | 1 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | ||
| χ6 | 4 | -4 | 1 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | ||
| χ7 | 4 | -4 | -2 | 0 | 2 | 0 | 0 | -1 | 1 | 0 | 0 | 0 |
| χ8 | 5 | 5 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | 1 | -1 | -1 |
| χ9 | 5 | 5 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 |
| χ10 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 0 | 0 |
| χ11 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | ||
| χ12 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 |
| maximal subgroup of | #91870, #91871, C2×#6728, … |
| maximal normal subgroup of | #91870, #91871, C2×#6728, … |
| maximal quotient of | #91143, C2×#6728, C3×#6728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(240, 89);