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Magma
magma: G := TransitiveGroup(38, 1);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{38}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $38$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,12,21,32,3,13,24,34,5,16,26,36,8,17,27,38,10,19,29,2,11,22,31,4,14,23,33,6,15,25,35,7,18,28,37,9,20,30) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $19$: $C_{19}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: $C_{19}$
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 |
| $1^{38}$ | $1$ | $1$ | $()$ | |
| $2^{19}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1, 3, 5, 8,10,11,14,15,18,20,21,24,26,27,29,31,33,35,37)( 2, 4, 6, 7, 9,12, 13,16,17,19,22,23,25,28,30,32,34,36,38)$ | |
| $38$ | $1$ | $38$ | $( 1, 4, 5, 7,10,12,14,16,18,19,21,23,26,28,29,32,33,36,37, 2, 3, 6, 8, 9,11, 13,15,17,20,22,24,25,27,30,31,34,35,38)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1, 5,10,14,18,21,26,29,33,37, 3, 8,11,15,20,24,27,31,35)( 2, 6, 9,13,17,22, 25,30,34,38, 4, 7,12,16,19,23,28,32,36)$ | |
| $38$ | $1$ | $38$ | $( 1, 6,10,13,18,22,26,30,33,38, 3, 7,11,16,20,23,27,32,35, 2, 5, 9,14,17,21, 25,29,34,37, 4, 8,12,15,19,24,28,31,36)$ | |
| $38$ | $1$ | $38$ | $( 1, 7,14,19,26,32,37, 6,11,17,24,30,35, 4,10,16,21,28,33, 2, 8,13,20,25,31, 38, 5,12,18,23,29,36, 3, 9,15,22,27,34)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1, 8,14,20,26,31,37, 5,11,18,24,29,35, 3,10,15,21,27,33)( 2, 7,13,19,25,32, 38, 6,12,17,23,30,36, 4, 9,16,22,28,34)$ | |
| $38$ | $1$ | $38$ | $( 1, 9,18,25,33, 4,11,19,27,36, 5,13,21,30,37, 7,15,23,31, 2,10,17,26,34, 3, 12,20,28,35, 6,14,22,29,38, 8,16,24,32)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,10,18,26,33, 3,11,20,27,35, 5,14,21,29,37, 8,15,24,31)( 2, 9,17,25,34, 4, 12,19,28,36, 6,13,22,30,38, 7,16,23,32)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,11,21,31, 3,14,24,33, 5,15,26,35, 8,18,27,37,10,20,29)( 2,12,22,32, 4,13, 23,34, 6,16,25,36, 7,17,28,38, 9,19,30)$ | |
| $38$ | $1$ | $38$ | $( 1,12,21,32, 3,13,24,34, 5,16,26,36, 8,17,27,38,10,19,29, 2,11,22,31, 4,14, 23,33, 6,15,25,35, 7,18,28,37, 9,20,30)$ | |
| $38$ | $1$ | $38$ | $( 1,13,26,38,11,23,35, 9,21,34, 8,19,31, 6,18,30, 3,16,27, 2,14,25,37,12,24, 36,10,22,33, 7,20,32, 5,17,29, 4,15,28)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,14,26,37,11,24,35,10,21,33, 8,20,31, 5,18,29, 3,15,27)( 2,13,25,38,12,23, 36, 9,22,34, 7,19,32, 6,17,30, 4,16,28)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,15,29, 5,20,33,10,24,37,14,27, 3,18,31, 8,21,35,11,26)( 2,16,30, 6,19,34, 9,23,38,13,28, 4,17,32, 7,22,36,12,25)$ | |
| $38$ | $1$ | $38$ | $( 1,16,29, 6,20,34,10,23,37,13,27, 4,18,32, 8,22,35,12,26, 2,15,30, 5,19,33, 9,24,38,14,28, 3,17,31, 7,21,36,11,25)$ | |
| $38$ | $1$ | $38$ | $( 1,17,33,12,27, 6,21,38,15,32,10,25, 3,19,35,13,29, 7,24, 2,18,34,11,28, 5, 22,37,16,31, 9,26, 4,20,36,14,30, 8,23)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,18,33,11,27, 5,21,37,15,31,10,26, 3,20,35,14,29, 8,24)( 2,17,34,12,28, 6, 22,38,16,32, 9,25, 4,19,36,13,30, 7,23)$ | |
| $38$ | $1$ | $38$ | $( 1,19,37,17,35,16,33,13,31,12,29, 9,27, 7,26, 6,24, 4,21, 2,20,38,18,36,15, 34,14,32,11,30,10,28, 8,25, 5,23, 3,22)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,20,37,18,35,15,33,14,31,11,29,10,27, 8,26, 5,24, 3,21)( 2,19,38,17,36,16, 34,13,32,12,30, 9,28, 7,25, 6,23, 4,22)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,21, 3,24, 5,26, 8,27,10,29,11,31,14,33,15,35,18,37,20)( 2,22, 4,23, 6,25, 7,28, 9,30,12,32,13,34,16,36,17,38,19)$ | |
| $38$ | $1$ | $38$ | $( 1,22, 3,23, 5,25, 8,28,10,30,11,32,14,34,15,36,18,38,20, 2,21, 4,24, 6,26, 7,27, 9,29,12,31,13,33,16,35,17,37,19)$ | |
| $38$ | $1$ | $38$ | $( 1,23, 8,30,14,36,20, 4,26, 9,31,16,37,22, 5,28,11,34,18, 2,24, 7,29,13,35, 19, 3,25,10,32,15,38,21, 6,27,12,33,17)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,24, 8,29,14,35,20, 3,26,10,31,15,37,21, 5,27,11,33,18)( 2,23, 7,30,13,36, 19, 4,25, 9,32,16,38,22, 6,28,12,34,17)$ | |
| $38$ | $1$ | $38$ | $( 1,25,11,36,21, 7,31,17, 3,28,14,38,24, 9,33,19, 5,30,15, 2,26,12,35,22, 8, 32,18, 4,27,13,37,23,10,34,20, 6,29,16)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,26,11,35,21, 8,31,18, 3,27,14,37,24,10,33,20, 5,29,15)( 2,25,12,36,22, 7, 32,17, 4,28,13,38,23, 9,34,19, 6,30,16)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,27,15, 3,29,18, 5,31,20, 8,33,21,10,35,24,11,37,26,14)( 2,28,16, 4,30,17, 6,32,19, 7,34,22, 9,36,23,12,38,25,13)$ | |
| $38$ | $1$ | $38$ | $( 1,28,15, 4,29,17, 5,32,20, 7,33,22,10,36,24,12,37,25,14, 2,27,16, 3,30,18, 6,31,19, 8,34,21, 9,35,23,11,38,26,13)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,29,20,10,37,27,18, 8,35,26,15, 5,33,24,14, 3,31,21,11)( 2,30,19, 9,38,28, 17, 7,36,25,16, 6,34,23,13, 4,32,22,12)$ | |
| $38$ | $1$ | $38$ | $( 1,30,20, 9,37,28,18, 7,35,25,15, 6,33,23,14, 4,31,22,11, 2,29,19,10,38,27, 17, 8,36,26,16, 5,34,24,13, 3,32,21,12)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,31,24,15, 8,37,29,21,14, 5,35,27,20,11, 3,33,26,18,10)( 2,32,23,16, 7,38, 30,22,13, 6,36,28,19,12, 4,34,25,17, 9)$ | |
| $38$ | $1$ | $38$ | $( 1,32,24,16, 8,38,29,22,14, 6,35,28,20,12, 3,34,26,17,10, 2,31,23,15, 7,37, 30,21,13, 5,36,27,19,11, 4,33,25,18, 9)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,33,27,21,15,10, 3,35,29,24,18,11, 5,37,31,26,20,14, 8)( 2,34,28,22,16, 9, 4,36,30,23,17,12, 6,38,32,25,19,13, 7)$ | |
| $38$ | $1$ | $38$ | $( 1,34,27,22,15, 9, 3,36,29,23,18,12, 5,38,31,25,20,13, 8, 2,33,28,21,16,10, 4,35,30,24,17,11, 6,37,32,26,19,14, 7)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,35,31,27,24,20,15,11, 8, 3,37,33,29,26,21,18,14,10, 5)( 2,36,32,28,23,19, 16,12, 7, 4,38,34,30,25,22,17,13, 9, 6)$ | |
| $38$ | $1$ | $38$ | $( 1,36,31,28,24,19,15,12, 8, 4,37,34,29,25,21,17,14, 9, 5, 2,35,32,27,23,20, 16,11, 7, 3,38,33,30,26,22,18,13,10, 6)$ | |
| $19^{2}$ | $1$ | $19$ | $( 1,37,35,33,31,29,27,26,24,21,20,18,15,14,11,10, 8, 5, 3)( 2,38,36,34,32,30, 28,25,23,22,19,17,16,13,12, 9, 7, 6, 4)$ | |
| $38$ | $1$ | $38$ | $( 1,38,35,34,31,30,27,25,24,22,20,17,15,13,11, 9, 8, 6, 3, 2,37,36,33,32,29, 28,26,23,21,19,18,16,14,12,10, 7, 5, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $38=2 \cdot 19$ | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 38.2 | magma: IdentifyGroup(G);
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| Character table: | 38 x 38 character table |
magma: CharacterTable(G);