Show commands:
Magma
magma: G := TransitiveGroup(25, 10);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_5:F_5$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,18,14,21)(2,20,13,24)(3,17,12,22)(4,19,11,25)(5,16,15,23)(6,9,10,7), (1,13,23,10,20)(2,14,24,6,16)(3,15,25,7,17)(4,11,21,8,18)(5,12,22,9,19) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$ x 2
Low degree siblings
10T10 x 2, 20T27 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,16,22,12)( 7,18,21,15)( 8,20,25,13)( 9,17,24,11)(10,19,23,14)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,12,22,16)( 7,15,21,18)( 8,13,25,20)( 9,11,24,17)(10,14,23,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)(14,19) (15,18)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 6,15,18,22)( 2, 7,11,19,23)( 3, 8,12,20,24)( 4, 9,13,16,25) ( 5,10,14,17,21)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,12,16,21)( 2, 8,13,17,22)( 3, 9,14,18,23)( 4,10,15,19,24) ( 5, 6,11,20,25)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 8,14,19,25)( 2, 9,15,20,21)( 3,10,11,16,22)( 4, 6,12,17,23) ( 5, 7,13,18,24)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 9,11,17,24)( 2,10,12,18,25)( 3, 6,13,19,21)( 4, 7,14,20,22) ( 5, 8,15,16,23)$ |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,10,13,20,23)( 2, 6,14,16,24)( 3, 7,15,17,25)( 4, 8,11,18,21) ( 5, 9,12,19,22)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $100=2^{2} \cdot 5^{2}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 100.12 | magma: IdentifyGroup(G);
|
Character table: |
2 2 2 2 2 . . . . . . 5 2 . . . 2 2 2 2 2 2 1a 4a 4b 2a 5a 5b 5c 5d 5e 5f 2P 1a 2a 2a 1a 5a 5c 5b 5f 5e 5d 3P 1a 4b 4a 2a 5a 5c 5b 5f 5e 5d 5P 1a 4a 4b 2a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 1 X.3 1 A -A -1 1 1 1 1 1 1 X.4 1 -A A -1 1 1 1 1 1 1 X.5 4 . . . 4 -1 -1 -1 -1 -1 X.6 4 . . . -1 -1 -1 -1 4 -1 X.7 4 . . . -1 B *B C -1 *C X.8 4 . . . -1 *B B *C -1 C X.9 4 . . . -1 C *C *B -1 B X.10 4 . . . -1 *C C B -1 *B A = -E(4) = -Sqrt(-1) = -i B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 C = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 = (3+Sqrt(5))/2 = 2+b5 |
magma: CharacterTable(G);