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Magma
magma: G := TransitiveGroup(26, 23);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:D_6$ | ||
| 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: | (1,11,12,3,6,5)(4,10,8,13,7,9)(14,15,25,21,20,23)(16,22,17,19,26,18), (1,20,12,16,10,25,8,21,6,17,4,26,2,22,13,18,11,14,9,23,7,19,5,15,3,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T34 x 2Siblings are shown 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 $ | $1$ | $1$ | $()$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,24,21,18,15,25,22,19,16,26,23,20,17)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,21,15,22,16,23,17,24,18,25,19,26,20)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ | |
| $ 13, 13 $ | $6$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ | |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $ 26 $ | $78$ | $26$ | $( 1,20,12,16,10,25, 8,21, 6,17, 4,26, 2,22,13,18,11,14, 9,23, 7,19, 5,15, 3,24 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,26, 5,21, 9,16,13,24, 4,19, 8,14,12,22, 3,17, 7,25,11,20, 2,15, 6,23,10,18 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,19,11,26, 8,20, 5,14, 2,21,12,15, 9,22, 6,16, 3,23,13,17,10,24, 7,18, 4,25 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,16, 8,17, 2,18, 9,19, 3,20,10,21, 4,22,11,23, 5,24,12,25, 6,26,13,14, 7,15 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,17, 9,20, 4,23,12,26, 7,16, 2,19,10,22, 5,25,13,15, 8,18, 3,21,11,24, 6,14 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,21,13,19,12,17,11,15,10,26, 9,24, 8,22, 7,20, 6,18, 5,16, 4,14, 3,25, 2,23 )$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $39$ | $2$ | $( 1,22)( 2,24)( 3,26)( 4,15)( 5,17)( 6,19)( 7,21)( 8,23)( 9,25)(10,14)(11,16) (12,18)(13,20)$ | |
| $ 26 $ | $78$ | $26$ | $( 1,22,10,20, 6,18, 2,16,11,14, 7,25, 3,23,12,21, 8,19, 4,17,13,15, 9,26, 5,24 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,15,12,14,10,26, 8,25, 6,24, 4,23, 2,22,13,21,11,20, 9,19, 7,18, 5,17, 3,16 )$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $39$ | $2$ | $( 1,21)( 2,15)( 3,22)( 4,16)( 5,23)( 6,17)( 7,24)( 8,18)( 9,25)(10,19)(11,26) (12,20)(13,14)$ | |
| $ 26 $ | $78$ | $26$ | $( 1,18,13,24,12,17,11,23,10,16, 9,22, 8,15, 7,21, 6,14, 5,20, 4,26, 3,19, 2,25 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,19, 9,23, 4,14,12,18, 7,22, 2,26,10,17, 5,21,13,25, 8,16, 3,20,11,24, 6,15 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,16, 8,26, 2,23, 9,20, 3,17,10,14, 4,24,11,21, 5,18,12,15, 6,25,13,22, 7,19 )$ | |
| $ 26 $ | $78$ | $26$ | $( 1,17, 4,25, 7,20,10,15,13,23, 3,18, 6,26, 9,21,12,16, 2,24, 5,19, 8,14,11,22 )$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2028=2^{2} \cdot 3 \cdot 13^{2}$ | 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: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 2028.s | magma: IdentifyGroup(G);
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| Character table: | 38 x 38 character table |
magma: CharacterTable(G);