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Magma
magma: G := TransitiveGroup(38, 30);
Group invariants
| Abstract group: | $D_{19}:F_{19}$ | magma: IdentifyGroup(G);
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| 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 | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,6)(2,5)(3,4)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(20,30)(21,29)(22,28)(23,27)(24,26)(31,38)(32,37)(33,36)(34,35)$, $(1,29,9,26,4,35,19,27,12,32,14,36,8,24,7,22,10,28)(2,31,6,20,13,34,11,30,17,23,18,25,15,38,5,37,16,21)(3,33)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ 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 $342$: $F_{19}$ x 2 $684$: 38T9 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: None
Low degree siblings
38T30 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{38}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{19}$ | $19$ | $2$ | $19$ | $( 1,24)( 2,34)( 3,25)( 4,35)( 5,26)( 6,36)( 7,27)( 8,37)( 9,28)(10,38)(11,29)(12,20)(13,30)(14,21)(15,31)(16,22)(17,32)(18,23)(19,33)$ |
| 2B | $2^{19}$ | $19$ | $2$ | $19$ | $( 1,36)( 2,26)( 3,35)( 4,25)( 5,34)( 6,24)( 7,33)( 8,23)( 9,32)(10,22)(11,31)(12,21)(13,30)(14,20)(15,29)(16,38)(17,28)(18,37)(19,27)$ |
| 2C | $2^{18},1^{2}$ | $361$ | $2$ | $18$ | $( 1, 6)( 2, 5)( 3, 4)( 7,19)( 8,18)( 9,17)(10,16)(11,15)(12,14)(20,30)(21,29)(22,28)(23,27)(24,26)(31,38)(32,37)(33,36)(34,35)$ |
| 3A1 | $3^{12},1^{2}$ | $361$ | $3$ | $24$ | $( 1,10,16)( 2,17, 8)( 3, 5,19)( 4,12,11)( 6, 7,14)(13,18,15)(20,28,27)(21,35,38)(22,23,30)(24,37,33)(26,32,36)(29,34,31)$ |
| 3A-1 | $3^{12},1^{2}$ | $361$ | $3$ | $24$ | $( 1,16,10)( 2, 8,17)( 3,19, 5)( 4,11,12)( 6,14, 7)(13,15,18)(20,27,28)(21,38,35)(22,30,23)(24,33,37)(26,36,32)(29,31,34)$ |
| 6A1 | $6^{6},1^{2}$ | $361$ | $6$ | $30$ | $( 1,16, 6,19, 4,14)( 2, 9,17,18,11, 3)( 5, 7,12,15,13, 8)(20,22,27,30,28,23)(21,34,38,29,35,31)(24,32,33,26,37,36)$ |
| 6A-1 | $6^{6},2$ | $361$ | $6$ | $31$ | $( 1,29,11,31, 5,26)( 2,33,18,21,16,32)( 3,37, 6,30, 8,38)( 4,22,13,20,19,25)( 7,34,15,28,14,24)( 9,23,10,27,17,36)(12,35)$ |
| 6B1 | $6^{6},1^{2}$ | $361$ | $6$ | $30$ | $( 1,10, 6,12, 3, 7)( 2,18,13,11,14,19)( 4,15, 8, 9,17, 5)(20,23,28,30,27,22)(21,31,35,29,38,34)(24,36,37,26,33,32)$ |
| 6B-1 | $6^{6},2$ | $361$ | $6$ | $31$ | $( 1,32)( 2,38,12,22, 8,36)( 3,25, 4,31,15,21)( 5,37, 7,30,10,29)( 6,24,18,20,17,33)( 9,23,13,28,19,26)(11,35,16,27,14,34)$ |
| 6C1 | $6^{6},2$ | $361$ | $6$ | $31$ | $( 1,37,19,24, 8,33)( 2,31,11,34,15,29)( 3,25)( 4,38,14,35,10,21)( 5,32, 6,26,17,36)( 7,20, 9,27,12,28)(13,22,18,30,16,23)$ |
| 6C-1 | $6^{6},2$ | $361$ | $6$ | $31$ | $( 1,21,10,23,16,37)( 2,36,17,33, 8,31)( 3,32, 5,24,19,25)( 4,28,12,34,11,38)( 6,20, 7,35,14,26)( 9,27)(13,30,18,29,15,22)$ |
| 9A1 | $9^{4},1^{2}$ | $361$ | $9$ | $32$ | $( 1,14,13,16, 7,15,10, 6,18)( 2,11, 3, 8,12,19,17, 4, 5)(20,21,37,27,38,24,28,35,33)(22,34,36,30,29,32,23,31,26)$ |
| 9A-1 | $9^{4},1^{2}$ | $361$ | $9$ | $32$ | $( 1, 7,18,16, 6,13,10,14,15)( 2,12, 5, 8, 4, 3,17,11,19)(20,38,33,27,35,37,28,21,24)(22,29,26,30,31,36,23,34,32)$ |
| 9A2 | $9^{4},1^{2}$ | $361$ | $9$ | $32$ | $( 1,13, 7,10,18,14,16,15, 6)( 2, 3,12,17, 5,11, 8,19, 4)(20,37,38,28,33,21,27,24,35)(22,36,29,23,26,34,30,32,31)$ |
| 9A-2 | $9^{4},1^{2}$ | $361$ | $9$ | $32$ | $( 1,15,14,10,13, 6,16,18, 7)( 2,19,11,17, 3, 4, 8, 5,12)(20,24,21,28,37,35,27,33,38)(22,32,34,23,36,31,30,26,29)$ |
| 9A4 | $9^{4},1^{2}$ | $361$ | $9$ | $32$ | $( 1, 6,15,16,14,18,10, 7,13)( 2, 4,19, 8,11, 5,17,12, 3)(20,35,24,27,21,33,28,38,37)(22,31,32,30,34,26,23,29,36)$ |
| 9A-4 | $9^{4},1^{2}$ | $361$ | $9$ | $32$ | $( 1,18, 6,10,15, 7,16,13,14)( 2, 5, 4,17,19,12, 8, 3,11)(20,33,35,28,24,38,27,37,21)(22,26,31,23,32,29,30,36,34)$ |
| 18A1 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,28, 6,32,15,24,16,21,14,27,18,34,10,20, 7,29,13,30)( 2,25, 4,38,19,31, 8,26,11,36, 5,35,17,37,12,33, 3,22)( 9,23)$ |
| 18A-1 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,34, 2,35, 6,20, 3,36,10,24,19,33,17,31, 9,23,15,29)( 4,37,14,28,16,30, 5,38,18,32,13,27,12,26, 8,22,11,25)( 7,21)$ |
| 18A5 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,35)( 2,34, 5,31,17,38, 8,28,10,26,18,37,12,24, 7,29, 6,30)( 3,33, 9,27,14,22,15,21,19,36,16,20, 4,32,13,23,11,25)$ |
| 18A-5 | $18^{2},1^{2}$ | $361$ | $18$ | $34$ | $( 1,19, 5,18,10,12, 2,14,11, 7, 8, 3, 9,17,15, 6,13,16)(20,31,33,23,35,32,28,29,24,30,38,36,27,34,37,22,21,26)$ |
| 18A7 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,24,14,20,16,34, 9,23, 5,33,19,36, 8,35,18,29, 2,31)( 3,38, 7,28,12,25, 4,26,13,32,10,30,11,37,17,22,15,27)( 6,21)$ |
| 18A-7 | $18^{2},1^{2}$ | $361$ | $18$ | $34$ | $( 1,12, 7,11, 4, 2,15,16,19, 9,17, 3,18, 6, 8,14,13,10)(20,29,37,23,38,26,28,34,33,30,21,32,27,31,24,22,35,36)$ |
| 18B1 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,29,18,20, 8,32,15,35,12,31,16,30,17,25, 3,38, 9,27)( 2,24, 4,33,14,21, 7,37,10,22, 6,23, 5,28,19,34,13,26)(11,36)$ |
| 18B-1 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,22, 5,34,16,29,13,20,19,38, 7,21,12,36, 2,25, 3,28)( 4,31,18,35, 9,27, 8,24,10,30, 6,37,14,23,17,32,11,33)(15,26)$ |
| 18B5 | $18^{2},1^{2}$ | $361$ | $18$ | $34$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(20,34,24,23,21,36,28,31,37,30,35,26,27,29,33,22,38,32)$ |
| 18B-5 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,21,19,35,14,29, 8,37,16,20,18,30, 9,23, 2,26, 5,22)( 3,31,10,28, 7,32,11,33,12,38,17,25, 4,36,15,34,13,24)( 6,27)$ |
| 18B7 | $18^{2},1^{2}$ | $361$ | $18$ | $34$ | $( 1, 8, 4, 9,17, 7,10,11, 5, 3,15,19,14, 6,16,13,12,18)(20,36,35,22,24,31,27,32,21,30,33,34,28,26,38,23,37,29)$ |
| 18B-7 | $18^{2},1^{2}$ | $361$ | $18$ | $34$ | $( 1,11, 9,17, 4,18,19,15,12, 5,14,16, 8, 2, 7, 6,10,13)(20,26,21,22,37,34,27,36,38,30,24,29,28,32,35,23,33,31)$ |
| 18C1 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,20,14,29,17,34, 6,22, 2,28, 4,25, 3,36,13,21, 8,38)( 5,33,12,32,18,23,15,37, 7,30,11,24, 9,27,10,35,19,31)(16,26)$ |
| 18C-1 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,26,13,37, 9,27, 4,24,12,25, 3,31, 6,29, 5,36,18,21)( 2,38,19,33, 7,22,11,32,16,35, 8,34,17,28,14,30,15,23)(10,20)$ |
| 18C5 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,30, 5,38,12,33,10,29,16,22,17,24,14,37, 4,36,15,20)( 2,32)( 3,34,18,26,11,31,13,35, 7,23, 6,21, 9,27,19,28, 8,25)$ |
| 18C-5 | $18^{2},1^{2}$ | $361$ | $18$ | $34$ | $( 1,13,19, 3,14,10, 8, 7,16,11,18,12, 9,17, 2, 4, 5,15)(20,32,38,22,33,29,27,26,35,30,37,31,28,36,21,23,24,34)$ |
| 18C7 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,20, 6,29,10,21,17,26,15,30, 2,37, 3,35,19,22, 9,23)( 4,33,16,28,18,24,12,36,11,38,14,32, 5,31,13,34, 8,25)( 7,27)$ |
| 18C-7 | $18^{2},2$ | $361$ | $18$ | $35$ | $( 1,30,15,32, 8,31, 2,22, 5,36,13,29, 9,23,11,26,10,34)( 3,33,14,21,18,27,16,24,17,35, 7,20,12,37,19,38, 6,28)( 4,25)$ |
| 19A | $19^{2}$ | $18$ | $19$ | $36$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| 19B | $19^{2}$ | $18$ | $19$ | $36$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| 19C | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,28,36,25,33,22,30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| 19D | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| 19E | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| 19F | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| 19G | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19H | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| 19I | $19,1^{19}$ | $36$ | $19$ | $18$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)$ |
| 19J | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 19K | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 38A | $38$ | $342$ | $38$ | $37$ | $( 1,23,18,22,16,21,14,20,12,38,10,37, 8,36, 6,35, 4,34, 2,33,19,32,17,31,15,30,13,29,11,28, 9,27, 7,26, 5,25, 3,24)$ |
| 38B | $38$ | $342$ | $38$ | $37$ | $( 1,27,19,37,18,28,17,38,16,29,15,20,14,30,13,21,12,31,11,22,10,32, 9,23, 8,33, 7,24, 6,34, 5,25, 4,35, 3,26, 2,36)$ |
Malle's constant $a(G)$: $1/18$
magma: ConjugacyClasses(G);
Character table
49 x 49 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed