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Magma
magma: G := TransitiveGroup(38, 38);
Group action invariants
Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{19}^2:C_9:C_{12}$ | ||
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: | (1,35,14,25,4,21,19,27,6,37,16,22)(2,24,3,32,11,20,18,38,17,30,9,23)(5,29,8,34,13,36,15,33,12,28,7,26)(10,31), (1,8,5,9,10,15,2,13,11)(3,18,17,12,6,14,16,7,19)(21,29,25,27,26,36,31,24,37)(22,38,30,34,32,33,23,28,35) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $S_3$, $C_6$ $12$: $C_{12}$, $C_3 : C_4$ $18$: $S_3\times C_3$, $D_{9}$ $36$: $C_3\times (C_3 : C_4)$, 36T9 $54$: 18T19 $108$: 36T68 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 $ | $108$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
$ 19, 19 $ | $108$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
$ 19, 19 $ | $108$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $19$ | $(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
$ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
$ 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 $ | $361$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $3$ | $(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
$ 19, 3, 3, 3, 3, 3, 3, 1 $ | $684$ | $57$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,25,22)(21,32,33) (23,27,36)(24,34,28)(26,29,31)(30,38,37)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ |
$ 19, 3, 3, 3, 3, 3, 3, 1 $ | $684$ | $57$ | $( 1,11, 7)( 2, 3,14)( 4, 6, 9)( 5,17,16)( 8,12,18)(10,15,13)(20,25,30,35,21, 26,31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
$ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $722$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,38)(22,37)(23,36) (24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
$ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $722$ | $6$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11) (21,32,31,38,27,28)(22,25,23,37,34,36)(24,30,26,35,29,33)$ |
$ 6, 6, 6, 6, 6, 6, 1, 1 $ | $722$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ |
$ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$ |
$ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$ |
$ 12, 12, 12, 2 $ | $3249$ | $12$ | $( 1,35,14,25, 4,21,19,27, 6,37,16,22)( 2,24, 3,32,11,20,18,38,17,30, 9,23) ( 5,29, 8,34,13,36,15,33,12,28, 7,26)(10,31)$ |
$ 12, 12, 12, 2 $ | $3249$ | $12$ | $( 1,30,19,31, 7,24,15,35,16,34, 9,22)( 2,29,12,38,18,32,14,36, 4,27,17,33) ( 3,28, 5,26,10,21,13,37,11,20, 6,25)( 8,23)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $3249$ | $4$ | $( 1,33, 8,22)( 2,26, 7,29)( 3,38, 6,36)( 4,31, 5,24)( 9,34,19,21)(10,27,18,28) (11,20,17,35)(12,32,16,23)(13,25,15,30)(14,37)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $3249$ | $4$ | $( 1,26,14,22)( 2,33,13,34)( 3,21,12,27)( 4,28,11,20)( 5,35,10,32)( 6,23, 9,25) ( 7,30, 8,37)(15,29,19,38)(16,36,18,31)(17,24)$ |
$ 12, 12, 12, 2 $ | $3249$ | $12$ | $( 1,24,18,21, 2,35, 7,33, 9,36, 6,22)( 3,27,15,26,16,37, 5,30,12,31,11,20) ( 4,38)( 8,25,17,29,13,23,19,32,10,28,14,34)$ |
$ 12, 12, 12, 2 $ | $3249$ | $12$ | $( 1,29, 9,37,10,38, 3,31,14,23,13,22)( 2,30)( 4,32, 7,35, 5,33,19,28,16,25,18, 27)( 6,34,12,21, 8,36,17,26,11,20,15,24)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $38988=2^{2} \cdot 3^{3} \cdot 19^{2}$ | 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: | 38988.o | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);