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Magma
magma: G := TransitiveGroup(38, 9);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times F_{19}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,32,6,25,33,22)(2,31,5,26,34,21)(3,9,19,24,18,7)(4,10,20,23,17,8)(11,35,37,16,29,28)(12,36,38,15,30,27)(13,14), (1,5,30,22,12,27,10,15,13,2,6,29,21,11,28,9,16,14)(3,17,26,36,19,37,31,34,8,4,18,25,35,20,38,32,33,7)(23,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $9$: $C_9$ $12$: $C_6\times C_2$ $18$: $C_{18}$ x 3 $36$: 36T2 $342$: $F_{19}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: $F_{19}$
Low degree siblings
38T9Siblings 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 $ | $18$ | $19$ | $( 1,16,30, 6,19,33,10,23,38,13,28, 3,18,31, 8,21,35,12,26)( 2,15,29, 5,20,34, 9,24,37,14,27, 4,17,32, 7,22,36,11,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $19$ | $3$ | $( 3,16,23)( 4,15,24)( 5,29, 7)( 6,30, 8)( 9,20,14)(10,19,13)(11,34,36) (12,33,35)(17,37,25)(18,38,26)(21,28,31)(22,27,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $19$ | $3$ | $( 3,23,16)( 4,24,15)( 5, 7,29)( 6, 8,30)( 9,14,20)(10,13,19)(11,36,34) (12,35,33)(17,25,37)(18,26,38)(21,31,28)(22,32,27)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1,32, 6,25,33,22)( 2,31, 5,26,34,21)( 3, 9,19,24,18, 7)( 4,10,20,23,17, 8) (11,35,37,16,29,28)(12,36,38,15,30,27)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $19$ | $2$ | $( 1,22)( 2,21)( 3,20)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,13)(10,14)(11,12) (23,37)(24,38)(25,35)(26,36)(27,33)(28,34)(29,31)(30,32)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1,27,35,17,30,22)( 2,28,36,18,29,21)( 3,14,19,15, 6,37)( 4,13,20,16, 5,38) ( 7,23,25,12,34,31)( 8,24,26,11,33,32)( 9,10)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $19$ | $9$ | $( 3,35,10,23,33,13,16,12,19)( 4,36, 9,24,34,14,15,11,20)( 5,32,17, 7,27,25,29, 22,37)( 6,31,18, 8,28,26,30,21,38)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $19$ | $9$ | $( 3,12,13,23,35,19,16,33,10)( 4,11,14,24,36,20,15,34, 9)( 5,22,25, 7,32,37,29, 27,17)( 6,21,26, 8,31,38,30,28,18)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $19$ | $9$ | $( 3,33,19,23,12,10,16,35,13)( 4,34,20,24,11, 9,15,36,14)( 5,27,37, 7,22,17,29, 32,25)( 6,28,38, 8,21,18,30,31,26)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,32, 8,11,23,22,16,36,19, 9,18, 4,38,25,28,34,13,29)( 2,31, 7,12,24,21,15, 35,20,10,17, 3,37,26,27,33,14,30)( 5, 6)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,22,23,27,35,14, 8,34,10,37,18,15,12, 4,26,32, 6,29)( 2,21,24,28,36,13, 7, 33, 9,38,17,16,11, 3,25,31, 5,30)(19,20)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,27,12,15,33,20,13, 5, 8,36,10,25,21, 4,18,24,31,29)( 2,28,11,16,34,19,14, 6, 7,35, 9,26,22, 3,17,23,32,30)(37,38)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $19$ | $9$ | $( 3,10,33,16,19,35,23,13,12)( 4, 9,34,15,20,36,24,14,11)( 5,17,27,29,37,32, 7, 25,22)( 6,18,28,30,38,31, 8,26,21)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $19$ | $9$ | $( 3,19,12,16,13,33,23,10,35)( 4,20,11,15,14,34,24, 9,36)( 5,37,22,29,25,27, 7, 17,32)( 6,38,21,30,26,28, 8,18,31)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $19$ | $9$ | $( 3,13,35,16,10,12,23,19,33)( 4,14,36,15, 9,11,24,20,34)( 5,25,32,29,17,22, 7, 37,27)( 6,26,31,30,18,21, 8,38,28)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,32, 3,20,38, 5, 8,34,30,15,23,14,35,17,12, 9,21,25)( 2,31, 4,19,37, 6, 7, 33,29,16,24,13,36,18,11,10,22,26)(27,28)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,22,18,34, 8,36,38,29,23, 9,28,32,16, 4,13,11,19,25)( 2,21,17,33, 7,35,37, 30,24,10,27,31,15, 3,14,12,20,26)( 5, 6)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,27,21,37, 8,11,13,34, 6,29, 3, 9,31,24,19,17,35,25)( 2,28,22,38, 7,12,14, 33, 5,30, 4,10,32,23,20,18,36,26)(15,16)$ |
| $ 18, 18, 1, 1 $ | $19$ | $18$ | $( 3,21,12,26,13, 8,23,31,35,38,19,30,16,28,33,18,10, 6)( 4,22,11,25,14, 7,24, 32,36,37,20,29,15,27,34,17, 9, 5)$ |
| $ 18, 18, 1, 1 $ | $19$ | $18$ | $( 3,28,35,26,10,30,23,21,33,38,13, 6,16,31,12,18,19, 8)( 4,27,36,25, 9,29,24, 22,34,37,14, 5,15,32,11,17,20, 7)$ |
| $ 18, 18, 1, 1 $ | $19$ | $18$ | $( 3,31,33,26,19, 6,23,28,12,38,10, 8,16,21,35,18,13,30)( 4,32,34,25,20, 5,24, 27,11,37, 9, 7,15,22,36,17,14,29)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,32,38,24, 6, 9,26,14, 3, 2,31,37,23, 5,10,25,13, 4)( 7,18,20,28,22,35,15, 12,34, 8,17,19,27,21,36,16,11,33)(29,30)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,22,12,36,23,29, 8,37, 3, 2,21,11,35,24,30, 7,38, 4)( 5,19,32,26, 9,18,14, 16,34, 6,20,31,25,10,17,13,15,33)(27,28)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,27,31,17,10,37,16,36, 3, 2,28,32,18, 9,38,15,35, 4)( 5,13,24, 8,25,19,22, 33,29, 6,14,23, 7,26,20,21,34,30)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $19$ | $2$ | $( 3,38)( 4,37)( 5,36)( 6,35)( 7,34)( 8,33)( 9,32)(10,31)(11,29)(12,30)(13,28) (14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $19$ | $6$ | $( 3,26,23,38,16,18)( 4,25,24,37,15,17)( 5,11, 7,36,29,34)( 6,12, 8,35,30,33) ( 9,22,14,32,20,27)(10,21,13,31,19,28)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $19$ | $6$ | $( 3,18,16,38,23,26)( 4,17,15,37,24,25)( 5,34,29,36, 7,11)( 6,33,30,35, 8,12) ( 9,27,20,32,14,22)(10,28,19,31,13,21)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1,32,19, 2,31,20)( 3,15,33, 4,16,34)( 5,38, 9, 6,37,10)( 7,21,24, 8,22,23) (11,28,14,12,27,13)(17,18)(25,30,36,26,29,35)$ |
| $ 38 $ | $18$ | $38$ | $( 1,22, 3,24, 6,25, 8,27,10,29,12,32,13,34,16,36,18,37,19, 2,21, 4,23, 5,26, 7,28, 9,30,11,31,14,33,15,35,17,38,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1,27,19, 2,28,20)( 3, 4)( 5,18,25, 6,17,26)( 7,31, 9, 8,32,10) (11,21,15,12,22,16)(13,36,38,14,35,37)(23,29,33,24,30,34)$ |
| $ 18, 18, 1, 1 $ | $19$ | $18$ | $( 3, 6,10,18,33,28,16,30,19,38,35,31,23, 8,13,26,12,21)( 4, 5, 9,17,34,27,15, 29,20,37,36,32,24, 7,14,25,11,22)$ |
| $ 18, 18, 1, 1 $ | $19$ | $18$ | $( 3,30,13,18,35,21,16, 8,10,38,12,28,23, 6,19,26,33,31)( 4,29,14,17,36,22,15, 7, 9,37,11,27,24, 5,20,25,34,32)$ |
| $ 18, 18, 1, 1 $ | $19$ | $18$ | $( 3, 8,19,18,12,31,16, 6,13,38,33,21,23,30,10,26,35,28)( 4, 7,20,17,11,32,15, 5,14,37,34,22,24,29, 9,25,36,27)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,32,18,22,10, 7,13,34,12, 2,31,17,21, 9, 8,14,33,11)( 3,25,35, 5,19,15,28, 29,23, 4,26,36, 6,20,16,27,30,24)(37,38)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,22,19,24,16,32,38,25,12, 2,21,20,23,15,31,37,26,11)( 3,17,28, 7,10, 5,13, 36,30, 4,18,27, 8, 9, 6,14,35,29)(33,34)$ |
| $ 18, 18, 2 $ | $19$ | $18$ | $( 1,27, 6, 9,30,15,21,14,12, 2,28, 5,10,29,16,22,13,11)( 3,37,18,32,26,34,35, 7,19, 4,38,17,31,25,33,36, 8,20)(23,24)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $684=2^{2} \cdot 3^{2} \cdot 19$ | 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: | 684.28 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);