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Magma
magma: G := TransitiveGroup(26, 49);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\SL(3,3):C_2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,22,8,25,5,23)(2,26,13,21,4,16)(3,24,9,17,11,19)(6,18)(7,20,10,14,12,15), (1,24,12,23,3,17,10,25,13,21,5,18)(2,20)(4,14,8,16,11,26)(6,15,7,19)(9,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $117$ | $2$ | $( 1,12)( 2, 6)( 3,10)( 5,13)(14,25)(15,19)(16,23)(18,26)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $104$ | $3$ | $( 2, 3, 5)( 4, 9, 8)( 6,10,13)(15,18,16)(17,21,22)(19,26,23)$ | |
| $ 6, 6, 3, 3, 2, 2, 1, 1, 1, 1 $ | $936$ | $6$ | $( 1,12)( 2,13, 3, 6, 5,10)( 4, 8, 9)(14,25)(15,23,18,19,16,26)(17,22,21)$ | |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2 $ | $234$ | $4$ | $( 1,14,12,25)( 2,19, 6,15)( 3,26,10,18)( 4,21)( 5,23,13,16)( 7,20)( 8,17) ( 9,22)(11,24)$ | |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ | $702$ | $4$ | $( 1, 5,12,13)( 2, 3, 6,10)( 4,11)( 8, 9)(14,23,25,16)(15,18,19,26)(17,22) (21,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $234$ | $2$ | $( 1,25)( 2,23)( 3,26)( 4,22)( 5,19)( 6,16)( 7,20)( 8,17)( 9,21)(10,18)(11,24) (12,14)(13,15)$ | |
| $ 12, 6, 4, 2, 2 $ | $936$ | $12$ | $( 1,14,12,25)( 2,23,10,15, 5,26, 6,16, 3,19,13,18)( 4,17, 9,21, 8,22)( 7,20) (11,24)$ | |
| $ 12, 6, 4, 2, 2 $ | $936$ | $12$ | $( 1,25,12,14)( 2,16,10,19, 5,18, 6,23, 3,15,13,26)( 4,17, 9,21, 8,22)( 7,20) (11,24)$ | |
| $ 8, 8, 4, 4, 2 $ | $1404$ | $8$ | $( 1,17,13,24, 8,15, 7,26)( 2,16,10,25,11,20, 9,22)( 3,21,12,18)( 4,19, 6,14) ( 5,23)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $624$ | $3$ | $( 1, 3, 5)( 2, 6, 9)( 4,12,10)( 8,13,11)(14,18,15)(16,23,19)(17,21,24) (20,26,25)$ | |
| $ 6, 6, 6, 6, 2 $ | $1872$ | $6$ | $( 1,19, 3,16, 5,23)( 2,21, 6,24, 9,17)( 4,18,12,15,10,14)( 7,22) ( 8,26,13,25,11,20)$ | |
| $ 13, 13 $ | $864$ | $13$ | $( 1, 2, 5, 9, 4, 3,10,11,12, 8, 6, 7,13)(14,19,20,23,17,25,24,26,18,22,21,15, 16)$ | |
| $ 13, 13 $ | $864$ | $13$ | $( 1,12, 9, 7,10, 2, 8, 4,13,11, 5, 6, 3)(14,18,23,15,24,19,22,17,16,26,20,21, 25)$ | |
| $ 8, 8, 4, 4, 1, 1 $ | $1404$ | $8$ | $( 1,13, 3, 9)( 2, 6, 5, 7, 8, 4,10,11)(14,22,18,21,17,20,16,25)(19,24,26,23)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $11232=2^{5} \cdot 3^{3} \cdot 13$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 11232.b | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);