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Magma
magma: G := TransitiveGroup(39, 33);
Group action invariants
| Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}:F_{13}$ | ||
| 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: | (1,2,7,6)(3,12,5,9)(8,11,13,10)(14,24,22,25)(15,16,21,20)(17,26,19,23)(27,38,28,30)(29,35,39,33)(31,32,37,36), (1,31,16,7,32,21,3,27,22,10,39,17)(2,29,19)(4,38,25,5,36,15,13,33,26,12,35,23)(6,34,18,8,30,24,11,37,20,9,28,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $12$: $C_{12}$ $52$: $C_{13}:C_4$ $156$: $F_{13}$, 39T10 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 13: None
Low degree siblings
39T33 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(27,37,34,31,28,38,35,32,29,39,36,33, 30)$ | |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(27,34,28,35,29,36,30,37,31,38,32,39, 33)$ | |
| $ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(27,28,29,30,31,32,33,34,35,36,37,38, 39)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,30,33,36,39,29,32,35,38,28,31,34,37)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,35,30,38,33,28,36,31,39,34,29,37,32)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,37,34,31,28,38,35,32,29,39,36,33,30)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,28,29,30,31,32,33,34,35,36,37,38,39)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,36,32,28,37,33,29,38,34,30,39,35,31)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,39,38,37,36,35,34,33,32,31,30,29,28)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,31,35,39,30,34,38,29,33,37,28,32,36)$ | |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,38,36,34,32,30,28,39,37,35,33,31,29)$ | |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,36,32,28,37,33,29,38,34,30,39,35,31)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,33,39,32,38,31,37,30,36,29,35,28,34)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,32,37,29,34,39,31,36,28,33,38,30,35)$ | |
| $ 13, 13, 13 $ | $12$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,35,30,38,33,28,36,31,39,34,29,37,32)$ | |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,19,24,16,21,26,18,23,15,20,25,17, 22)(27,32,37,29,34,39,31,36,28,33,38,30,35)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)(27,33)(28,32)(29,31)(34,39)(35,38)(36,37)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20) (27,28,33,32)(29,38,31,35)(34,37,39,36)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $169$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21) (27,32,33,28)(29,35,31,38)(34,36,39,37)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $13$ | $3$ | $( 1,32,22)( 2,29,19)( 3,39,16)( 4,36,26)( 5,33,23)( 6,30,20)( 7,27,17) ( 8,37,14)( 9,34,24)(10,31,21)(11,28,18)(12,38,15)(13,35,25)$ | |
| $ 39 $ | $52$ | $39$ | $( 1,29,19,12,35,25,10,28,18, 8,34,24, 6,27,17, 4,33,23, 2,39,16,13,32,22,11, 38,15, 9,31,21, 7,37,14, 5,30,20, 3,36,26)$ | |
| $ 39 $ | $52$ | $39$ | $( 1,39,16,10,38,15, 6,37,14, 2,36,26,11,35,25, 7,34,24, 3,33,23,12,32,22, 8, 31,21, 4,30,20,13,29,19, 9,28,18, 5,27,17)$ | |
| $ 39 $ | $52$ | $39$ | $( 1,33,23, 6,31,21,11,29,19, 3,27,17, 8,38,15,13,36,26, 5,34,24,10,32,22, 2, 30,20, 7,28,18,12,39,16, 4,37,14, 9,35,25)$ | |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,28,23,10,29,22)( 2,31,20, 9,39,25)( 3,34,17, 8,36,15)( 4,37,14, 7,33,18) ( 5,27,24, 6,30,21)(11,32,19,13,38,26)(12,35,16)$ | |
| $ 12, 12, 12, 3 $ | $169$ | $12$ | $( 1,27,16,11,33,20,13,29,26, 3,36,22)( 2,38,19, 6,30,18,12,31,23, 8,39,24) ( 4,34,25, 9,37,14,10,35,17, 5,32,15)( 7,28,21)$ | |
| $ 12, 12, 12, 3 $ | $169$ | $12$ | $( 1,33,21, 8,34,16, 4,39,17,10,38,22)( 2,35,24,13,31,18, 3,37,14, 5,28,20) ( 6,30,23, 7,32,26,12,29,15,11,27,25)( 9,36,19)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $13$ | $3$ | $( 1,22,32)( 2,19,29)( 3,16,39)( 4,26,36)( 5,23,33)( 6,20,30)( 7,17,27) ( 8,14,37)( 9,24,34)(10,21,31)(11,18,28)(12,15,38)(13,25,35)$ | |
| $ 39 $ | $52$ | $39$ | $( 1,22,29,12,15,35,10,21,28, 8,14,34, 6,20,27, 4,26,33, 2,19,39,13,25,32,11, 18,38, 9,24,31, 7,17,37, 5,23,30, 3,16,36)$ | |
| $ 39 $ | $52$ | $39$ | $( 1,22,39,10,21,38, 6,20,37, 2,19,36,11,18,35, 7,17,34, 3,16,33,12,15,32, 8, 14,31, 4,26,30,13,25,29, 9,24,28, 5,23,27)$ | |
| $ 39 $ | $52$ | $39$ | $( 1,22,33, 6,20,31,11,18,29, 3,16,27, 8,14,38,13,25,36, 5,23,34,10,21,32, 2, 19,30, 7,17,28,12,15,39, 4,26,37, 9,24,35)$ | |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $169$ | $6$ | $( 1,19,31, 5,18,32)( 2,22,28, 4,15,35)( 3,25,38)( 6,21,29,13,16,34) ( 7,24,39,12,26,37)( 8,14,36,11,23,27)( 9,17,33,10,20,30)$ | |
| $ 12, 12, 12, 3 $ | $169$ | $12$ | $( 1,15,31, 7,16,36, 3,24,37,10,23,32)( 2,26,34)( 4,22,27, 5,20,30,13,17,28,12, 19,38)( 6,18,33, 8,14,39,11,21,35, 9,25,29)$ | |
| $ 12, 12, 12, 3 $ | $169$ | $12$ | $( 1,26,39, 4,19,35, 6,23,28, 3,17,32)( 2,15,29, 9,16,37, 5,21,38,11,20,30) ( 7,25,31, 8,14,34,13,24,36,12,22,33)(10,18,27)$ |
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.v | magma: IdentifyGroup(G);
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| Character table: | 34 x 34 character table |
magma: CharacterTable(G);