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Magma
magma: G := TransitiveGroup(38, 25);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{19}^2:D_{18}$ | ||
| 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: | (1,36,15,27,10,37,5,28,19,38,14,29,9,20,4,30,18,21,13,31,8,22,3,32,17,23,12,33,7,24,2,34,16,25,11,35,6,26), (1,30,2,26,3,22,4,37,5,33,6,29,7,25,8,21,9,36,10,32,11,28,12,24,13,20,14,35,15,31,16,27,17,23,18,38,19,34) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $18$: $D_{9}$ $36$: $D_{18}$ 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 $ | $18$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $19$ | $(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ |
| $ 38 $ | $342$ | $38$ | $( 1,36,15,27,10,37, 5,28,19,38,14,29, 9,20, 4,30,18,21,13,31, 8,22, 3,32,17, 23,12,33, 7,24, 2,34,16,25,11,35, 6,26)$ |
| $ 38 $ | $342$ | $38$ | $( 1,27, 5,38, 9,30,13,22,17,33, 2,25, 6,36,10,28,14,20,18,31, 3,23, 7,34,11, 26,15,37,19,29, 4,21, 8,32,12,24,16,35)$ |
| $ 38 $ | $342$ | $38$ | $( 1,37,14,30, 8,23, 2,35,15,28, 9,21, 3,33,16,26,10,38, 4,31,17,24,11,36, 5, 29,18,22,12,34, 6,27,19,20,13,32, 7,25)$ |
| $ 38 $ | $342$ | $38$ | $( 1,38,13,33, 6,28,18,23,11,37, 4,32,16,27, 9,22, 2,36,14,31, 7,26,19,21,12, 35, 5,30,17,25,10,20, 3,34,15,29, 8,24)$ |
| $ 38 $ | $342$ | $38$ | $( 1,21,11,20, 2,38,12,37, 3,36,13,35, 4,34,14,33, 5,32,15,31, 6,30,16,29, 7, 28,17,27, 8,26,18,25, 9,24,19,23,10,22)$ |
| $ 38 $ | $342$ | $38$ | $( 1,33,18,37,16,22,14,26,12,30,10,34, 8,38, 6,23, 4,27, 2,31,19,35,17,20,15, 24,13,28,11,32, 9,36, 7,21, 5,25, 3,29)$ |
| $ 38 $ | $342$ | $38$ | $( 1,30, 2,28, 3,26, 4,24, 5,22, 6,20, 7,37, 8,35, 9,33,10,31,11,29,12,27,13, 25,14,23,15,21,16,38,17,36,18,34,19,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $171$ | $2$ | $( 1,31)( 2,29)( 3,27)( 4,25)( 5,23)( 6,21)( 7,38)( 8,36)( 9,34)(10,32)(11,30) (12,28)(13,26)(14,24)(15,22)(16,20)(17,37)(18,35)(19,33)$ |
| $ 38 $ | $342$ | $38$ | $( 1,34,17,21,14,27,11,33, 8,20, 5,26, 2,32,18,38,15,25,12,31, 9,37, 6,24, 3, 30,19,36,16,23,13,29,10,35, 7,22, 4,28)$ |
| $ 38 $ | $342$ | $38$ | $( 1,20,12,36, 4,33,15,30, 7,27,18,24,10,21, 2,37,13,34, 5,31,16,28, 8,25,19, 22,11,38, 3,35,14,32, 6,29,17,26, 9,23)$ |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(21,30,25,32,26,23,31, 35,37,38,29,34,27,33,36,28,24,22)$ |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)(21,33,37,32,24,34,31, 30,36,38,26,22,27,35,25,28,29,23)$ |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)(21,35,36,32,29,22,31, 33,25,38,24,23,27,30,37,28,26,34)$ |
| $ 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 $ | $722$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 38 $ | $342$ | $38$ | $( 1,36,19,21,18,25,17,29,16,33,15,37,14,22,13,26,12,30,11,34,10,38, 9,23, 8, 27, 7,31, 6,35, 5,20, 4,24, 3,28, 2,32)$ |
| $ 38 $ | $342$ | $38$ | $( 1,37, 4,25, 7,32,10,20,13,27,16,34,19,22, 3,29, 6,36, 9,24,12,31,15,38,18, 26, 2,33, 5,21, 8,28,11,35,14,23,17,30)$ |
| $ 38 $ | $342$ | $38$ | $( 1,38, 8,29,15,20, 3,30,10,21,17,31, 5,22,12,32,19,23, 7,33,14,24, 2,34, 9, 25,16,35, 4,26,11,36,18,27, 6,37,13,28)$ |
| $ 38 $ | $342$ | $38$ | $( 1,21,16,37,12,34, 8,31, 4,28,19,25,15,22,11,38, 7,35, 3,32,18,29,14,26,10, 23, 6,20, 2,36,17,33,13,30, 9,27, 5,24)$ |
| $ 38 $ | $342$ | $38$ | $( 1,33, 7,28,13,23,19,37, 6,32,12,27,18,22, 5,36,11,31,17,26, 4,21,10,35,16, 30, 3,25, 9,20,15,34, 2,29, 8,24,14,38)$ |
| $ 38 $ | $342$ | $38$ | $( 1,24, 9,30,17,36, 6,23,14,29, 3,35,11,22,19,28, 8,34,16,21, 5,27,13,33, 2, 20,10,26,18,32, 7,38,15,25, 4,31,12,37)$ |
| $ 38 $ | $342$ | $38$ | $( 1,31,18,20,16,28,14,36,12,25,10,33, 8,22, 6,30, 4,38, 2,27,19,35,17,24,15, 32,13,21,11,29, 9,37, 7,26, 5,34, 3,23)$ |
| $ 38 $ | $342$ | $38$ | $( 1,35,15,36,10,37, 5,38,19,20,14,21, 9,22, 4,23,18,24,13,25, 8,26, 3,27,17, 28,12,29, 7,30, 2,31,16,32,11,33, 6,34)$ |
| $ 38 $ | $342$ | $38$ | $( 1,34,11,32, 2,30,12,28, 3,26,13,24, 4,22,14,20, 5,37,15,35, 6,33,16,31, 7, 29,17,27, 8,25,18,23, 9,21,19,38,10,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $171$ | $2$ | $( 1,22)( 2,37)( 3,33)( 4,29)( 5,25)( 6,21)( 7,36)( 8,32)( 9,28)(10,24)(11,20) (12,35)(13,31)(14,27)(15,23)(16,38)(17,34)(18,30)(19,26)$ |
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.z | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);