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Magma
magma: G := TransitiveGroup(38, 17);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{19}^2:C_3:C_4$ | ||
| 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));
| |
| Generators: | (1,38,12,25)(2,23,11,21)(3,27,10,36)(4,31,9,32)(5,35,8,28)(6,20,7,24)(13,29,19,34)(14,33,18,30)(15,37,17,26)(16,22), (1,22,13,35)(2,31,12,26)(3,21,11,36)(4,30,10,27)(5,20,9,37)(6,29,8,28)(7,38)(14,25,19,32)(15,34,18,23)(16,24,17,33) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ $12$: $C_3 : C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: 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
| 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 $ | $1$ | $1$ | $()$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $361$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $722$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1083$ | $4$ | $( 1,38,12,25)( 2,23,11,21)( 3,27,10,36)( 4,31, 9,32)( 5,35, 8,28)( 6,20, 7,24) (13,29,19,34)(14,33,18,30)(15,37,17,26)(16,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1083$ | $4$ | $( 1,21,19,25)( 2,36,18,29)( 3,32,17,33)( 4,28,16,37)( 5,24,15,22)( 6,20,14,26) ( 7,35,13,30)( 8,31,12,34)( 9,27,11,38)(10,23)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $4332=2^{2} \cdot 3 \cdot 19^{2}$ | 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: | 4332.o | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);