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Magma
magma: G := TransitiveGroup(38, 26);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{19}^2:C_9:C_4$ | ||
| 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));
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| Generators: | (1,32,12,31)(2,25,11,38)(3,37,10,26)(4,30,9,33)(5,23,8,21)(6,35,7,28)(13,24,19,20)(14,36,18,27)(15,29,17,34)(16,22), (1,4,12,8,10,9,19,14,7)(2,13,17,15,16,6,11,18,5)(20,22,37,26,29,23,35,30,21)(24,33,34,32,36,28,25,31,38) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ $12$: $C_3 : C_4$ $18$: $D_{9}$ $36$: 36T9 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 $ | $36$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 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 $ | $36$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,38,37,36,35,34, 33,32,31,30,29,28,27,26,25,24,23,22,21)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $19$ | $(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| $ 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 $ | $722$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $722$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)(21,33,37,32,24,34,31, 30,36,38,26,22,27,35,25,28,29,23)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)(21,35,36,32,29,22,31, 33,25,38,24,23,27,30,37,28,26,34)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(21,30,25,32,26,23,31, 35,37,38,29,34,27,33,36,28,24,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $3249$ | $4$ | $( 1,32,12,31)( 2,25,11,38)( 3,37,10,26)( 4,30, 9,33)( 5,23, 8,21)( 6,35, 7,28) (13,24,19,20)(14,36,18,27)(15,29,17,34)(16,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $3249$ | $4$ | $( 1,27, 7,31)( 2,34, 6,24)( 3,22, 5,36)( 4,29)( 8,38,19,20)( 9,26,18,32) (10,33,17,25)(11,21,16,37)(12,28,15,30)(13,35,14,23)$ |
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.ba | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);