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Magma
magma: G := TransitiveGroup(38, 29);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{19}^2:C_{36}$ | ||
| 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: | (1,35,2,23,15,38,13,24,6,32,10,22,5,25,16,26,7,20,4,37,3,30,9,34,11,29,18,21,14,31,19,28,8,27,17,33)(12,36), (1,15,5,4,2,17,9,12,18,11,16,7,8,10,14,3,19,13)(20,32,37,28,29,31,35,24,21,34,22,36,26,25,23,38,30,33) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $9$: $C_9$ $12$: $C_{12}$ $18$: $C_{18}$ $36$: $C_{36}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: None
Low degree siblings
38T29 x 9Siblings are shown 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 $ | $36$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $19$ | $(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,33,37,32,24,34,31, 30,36,38,26,22,27,35,25,28,29,23)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,30,25,32,26,23,31, 35,37,38,29,34,27,33,36,28,24,22)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,35,36,32,29,22,31, 33,25,38,24,23,27,30,37,28,26,34)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $361$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,35, 2,23,15,38,13,24, 6,32,10,22, 5,25,16,26, 7,20, 4,37, 3,30, 9,34,11, 29,18,21,14,31,19,28, 8,27,17,33)(12,36)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,33)( 2,34,16,29,17,30,13,26,10,23, 3,35,12,25,14,27, 6,38,19,32, 5,37, 4, 36, 8,21,11,24,18,31, 9,22, 7,20,15,28)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,30,19,38, 9,23, 4,25,11,26, 5,36, 2,22,10,34,14,21,16,24,17,35, 8,31,13, 29, 6,28,12,37,15,32, 7,20, 3,33)(18,27)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,28, 6,34,19,23,11,21,13,31, 3,38,15,22,12,26, 8,25, 9,30, 4,24,10,35,18, 37,16,27, 7,20,14,36,17,32, 2,33)( 5,29)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,32,12,29, 7,20,11,31, 4,26, 2,30,15,23,16,21,19,34, 9,35,17,38, 3,28,18, 36, 6,22, 8,37,14,25,13,27,10,33)( 5,24)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,38, 4,29,10,30, 3,32, 8,36,18,25,19,22, 2,35, 6,23,14,37,11,27, 5,26,12, 24, 7,20,16,31,15,34,13,21, 9,33)(17,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $361$ | $4$ | $( 1,23,19,33)( 2,32,18,24)( 3,22,17,34)( 4,31,16,25)( 5,21,15,35)( 6,30,14,26) ( 7,20,13,36)( 8,29,12,27)( 9,38,11,37)(10,28)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,34, 4,27, 2,38,16,37,13,25,15,33)( 3,23, 9,28, 5,31,14,29, 8,24,12,21) ( 6,35, 7,20,19,30,11,36,10,32,17,22)(18,26)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,22,18,29, 2,28, 7,20, 9,32, 6,33)( 3,34,15,30,16,36, 5,27,12,31,11,25) ( 4,21)( 8,26,17,23,13,37,19,35,10,38,14,24)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,25, 6,24, 9,31, 7,20, 2,21,18,33)( 3,36,11,23,12,38, 5,28,16,22,15,26) ( 4,32)( 8,35,14,30,10,27,19,29,13,34,17,37)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,37, 2,31,10,21,17,36,16,23, 8,33)( 3,25,18,30, 5,32,15,29,19,24,13,22) ( 4,38, 7,20,12,28,14,35,11,34, 6,26)( 9,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $361$ | $4$ | $( 1,36,14,33)( 2,27,13,23)( 3,37,12,32)( 4,28,11,22)( 5,38,10,31)( 6,29, 9,21) ( 7,20, 8,30)(15,24,19,26)(16,34,18,35)(17,25)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,29,15,27, 3,26,16,35,13,30, 2,37, 6,31, 8,28, 9,36,19,21, 5,23,17,24, 4, 34, 7,20,18,32,14,38,12,22,11,33)(10,25)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,24,15,21, 7,20,17,26,14,28,13,35,19,31, 2,36, 9,25, 5,34,10,37,18,38, 8, 32,11,30,12,23, 6,27, 4,22,16,33)( 3,29)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,26,11,35, 9,37,17,29, 4,23,18,28,19,27,15,31,12,34, 5,22,14,32,16,30, 8, 38, 2,25, 7,20, 6,21,10,36,13,33)( 3,24)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,21,12,35,15,25, 2,24,14,22,19,37,10,29,11,32,13,38,17,31, 6,36, 3,27,16, 28, 4,30,18,34, 8,23, 7,20, 5,33)( 9,26)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,31,18,22, 9,29,16,32,19,36, 4,35, 3,21, 8,34, 2,26,13,28,15,37, 5,30,17, 27,14,23,10,24,11,38, 6,25,12,33)( 7,20)$ |
| $ 36, 2 $ | $361$ | $36$ | $( 1,27,11,28, 3,31,17,21, 2,29,14,34,12,30, 6,37, 7,20,10,26,19,25, 8,22,13, 32, 9,24,16,38,18,23, 5,35, 4,33)(15,36)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$ | 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: | 12996.p | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);