Show commands:
Magma
magma: G := TransitiveGroup(28, 32);
Group action invariants
Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $\PSL(2,7)$ | ||
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: | (1,13,28,18,22,6,10)(2,14,26,17,24,7,12)(3,16,25,19,23,8,11)(4,15,27,20,21,5,9), (1,14)(2,15)(3,16)(4,13)(5,28)(6,26)(7,25)(8,27)(11,12)(17,18)(19,20)(21,24) | magma: Generators(G);
|
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 7: $\GL(3,2)$ x 2
Degree 14: None
Low degree siblings
7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 42T37, 42T38 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, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $21$ | $2$ | $( 3, 4)( 5,26)( 6,28)( 7,25)( 8,27)( 9,22)(10,23)(11,24)(12,21)(13,16)(17,19) (18,20)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $56$ | $3$ | $( 2, 3, 4)( 5,13,25)( 6,15,28)( 7,16,26)( 8,14,27)( 9,24,20)(10,21,19) (11,22,18)(12,23,17)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2 $ | $42$ | $4$ | $( 1, 2, 3, 4)( 5,16,24,18)( 6,14,23,17)( 7,13,22,20)( 8,15,21,19)( 9,26) (10,28,12,27)(11,25)$ |
$ 7, 7, 7, 7 $ | $24$ | $7$ | $( 1, 5,13,17,25,23,12)( 2, 6,14,18,26,21, 9)( 3, 7,15,19,27,24,11) ( 4, 8,16,20,28,22,10)$ |
$ 7, 7, 7, 7 $ | $24$ | $7$ | $( 1, 5,15,22,11,28,19)( 2, 7,13,21,12,27,18)( 3, 6,14,24,10,25,20) ( 4, 8,16,23, 9,26,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $168=2^{3} \cdot 3 \cdot 7$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | no | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 168.42 | magma: IdentifyGroup(G);
|
Character table: |
2 3 3 . 2 . . 3 1 . 1 . . . 7 1 . . . 1 1 1a 2a 3a 4a 7a 7b 2P 1a 1a 3a 2a 7a 7b 3P 1a 2a 1a 4a 7b 7a 5P 1a 2a 3a 4a 7b 7a 7P 1a 2a 3a 4a 1a 1a X.1 1 1 1 1 1 1 X.2 3 -1 . 1 A /A X.3 3 -1 . 1 /A A X.4 6 2 . . -1 -1 X.5 7 -1 1 -1 . . X.6 8 . -1 . 1 1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |
magma: CharacterTable(G);