Show commands:
Magma
magma: G := TransitiveGroup(25, 41);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $41$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_5^2:\SL(2,3)$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,25,10,21,2,17)(3,14,22,24,13,5)(4,6,18,23,16,9)(7,15,19,20,12,8), (1,25,13,19)(2,11,12,3)(4,18,15,21)(5,9,14,10)(6,16,8,23)(17,24,22,20) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $12$: $A_4$ $24$: $\SL(2,3)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Low degree siblings
30T126Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 5 $ | $24$ | $5$ | $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6) ( 5,23,16,14, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $150$ | $4$ | $( 2, 4, 5, 3)( 6,14,21,18)( 7,12,25,20)( 8,15,24,17)( 9,13,23,19)(10,11,22,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $100$ | $3$ | $( 2,22, 9)( 3,18,12)( 4,14,20)( 5,10,23)( 6,19, 8)( 7,15,11)(13,24,21) (16,25,17)$ |
$ 6, 6, 6, 6, 1 $ | $100$ | $6$ | $( 2,10, 9, 5,22,23)( 3,14,12, 4,18,20)( 6,13, 8,21,19,24)( 7,17,11,25,15,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $100$ | $3$ | $( 2, 9,22)( 3,12,18)( 4,20,14)( 5,23,10)( 6, 8,19)( 7,11,15)(13,21,24) (16,17,25)$ |
$ 6, 6, 6, 6, 1 $ | $100$ | $6$ | $( 2,23,22, 5, 9,10)( 3,20,18, 4,12,14)( 6,24,19,21, 8,13)( 7,16,15,25,11,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $600=2^{3} \cdot 3 \cdot 5^{2}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 600.150 | magma: IdentifyGroup(G);
|
Character table: |
2 3 . 3 2 1 1 1 1 3 1 . 1 . 1 1 1 1 5 2 2 . . . . . . 1a 5a 2a 4a 3a 6a 3b 6b 2P 1a 5a 1a 2a 3b 3b 3a 3a 3P 1a 5a 2a 4a 1a 2a 1a 2a 5P 1a 1a 2a 4a 3b 6b 3a 6a X.1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 A A /A /A X.3 1 1 1 1 /A /A A A X.4 2 2 -2 . -1 1 -1 1 X.5 2 2 -2 . -A A -/A /A X.6 2 2 -2 . -/A /A -A A X.7 3 3 3 -1 . . . . X.8 24 -1 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 |
magma: CharacterTable(G);