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Magma
magma: G := TransitiveGroup(38, 24);
Group invariants
| Abstract group: | $D_{19}^2:C_6$ | magma: IdentifyGroup(G);
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| Order: | $8664=2^{3} \cdot 3 \cdot 19^{2}$ | 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 | 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 $t$: | $24$ | 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: | $(1,26,7,38,17,20,2,28,15,35,5,34)(3,30,4,32,12,29,19,24,18,22,10,25)(6,36,9,23,14,33,16,37,13,31,8,21)(11,27)$, $(1,10,16)(2,17,8)(3,5,19)(4,12,11)(6,7,14)(13,18,15)(20,27,35,36,29,21)(22,32,38,34,24,37)(23,25,30,33,31,26)$ | 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 $8$: $D_{4}$ $12$: $C_6\times C_2$ $24$: $D_4 \times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: None
Low degree siblings
38T24Siblings 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 | $2^{9},1^{20}$ | $38$ | $2$ | $9$ | $( 1, 3)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)$ |
| 2B | $2^{19}$ | $38$ | $2$ | $19$ | $( 1,32)( 2,29)( 3,26)( 4,23)( 5,20)( 6,36)( 7,33)( 8,30)( 9,27)(10,24)(11,21)(12,37)(13,34)(14,31)(15,28)(16,25)(17,22)(18,38)(19,35)$ |
| 2C | $2^{18},1^{2}$ | $361$ | $2$ | $18$ | $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(35,38)(36,37)$ |
| 3A1 | $3^{12},1^{2}$ | $361$ | $3$ | $24$ | $( 1, 4, 6)( 2,11,17)( 3,18, 9)( 5,13,12)( 7, 8,15)(14,19,16)(20,34,37)(21,22,29)(23,36,32)(25,31,35)(26,38,27)(28,33,30)$ |
| 3A-1 | $3^{12},1^{2}$ | $361$ | $3$ | $24$ | $( 1, 6, 4)( 2,17,11)( 3, 9,18)( 5,12,13)( 7,15, 8)(14,16,19)(20,37,34)(21,29,22)(23,32,36)(25,35,31)(26,27,38)(28,30,33)$ |
| 4A | $4^{9},2$ | $722$ | $4$ | $28$ | $( 1,30,18,24)( 2,33,17,21)( 3,36,16,37)( 4,20,15,34)( 5,23,14,31)( 6,26,13,28)( 7,29,12,25)( 8,32,11,22)( 9,35,10,38)(19,27)$ |
| 6A1 | $6^{6},1^{2}$ | $361$ | $6$ | $30$ | $( 1,12,11,18, 7, 8)( 2, 5, 3,17,14,16)( 4,10, 6,15, 9,13)(20,38,26,34,35,28)(21,31,37,33,23,36)(22,24,29,32,30,25)$ |
| 6A-1 | $6^{6},1^{2}$ | $361$ | $6$ | $30$ | $( 1, 8, 7,18,11,12)( 2,16,14,17, 3, 5)( 4,13, 9,15, 6,10)(20,28,35,34,26,38)(21,36,23,33,37,31)(22,25,30,32,29,24)$ |
| 6B1 | $6^{6},2$ | $722$ | $6$ | $31$ | $( 1,36, 4,32, 6,23)( 2,22,11,29,17,21)( 3,27,18,26, 9,38)( 5,37,13,20,12,34)( 7,28, 8,33,15,30)(10,24)(14,25,19,31,16,35)$ |
| 6B-1 | $6^{6},2$ | $722$ | $6$ | $31$ | $( 1,23, 6,32, 4,36)( 2,21,17,29,11,22)( 3,38, 9,26,18,27)( 5,34,12,20,13,37)( 7,30,15,33, 8,28)(10,24)(14,35,16,31,19,25)$ |
| 6C1 | $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $27$ | $( 1,15,13,16, 2, 4)( 3,12, 8,14, 5, 9)( 6,17,10,11,19, 7)(20,31,38)(21,23,26)(22,34,33)(24,37,28)(25,29,35)(27,32,30)$ |
| 6C-1 | $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $27$ | $( 1, 4, 2,16,13,15)( 3, 9, 5,14, 8,12)( 6, 7,19,11,10,17)(20,38,31)(21,26,23)(22,33,34)(24,28,37)(25,35,29)(27,30,32)$ |
| 12A1 | $12^{3},2$ | $722$ | $12$ | $34$ | $( 1,32,12,30,11,25,18,22, 7,24, 8,29)( 2,37, 5,33, 3,23,17,36,14,21,16,31)( 4,28,10,20, 6,38,15,26, 9,34,13,35)(19,27)$ |
| 12A-1 | $12^{3},2$ | $722$ | $12$ | $34$ | $( 1,29, 8,24, 7,22,18,25,11,30,12,32)( 2,31,16,21,14,36,17,23, 3,33, 5,37)( 4,35,13,34, 9,26,15,38, 6,20,10,28)(19,27)$ |
| 19A1 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 19A2 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,26,32,38,25,31,37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| 19A4 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| 19B1 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| 19B2 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| 19B4 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 19C1 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| 19C2 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| 19C4 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| 19D1 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,26,32,38,25,31,37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| 19D2 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| 19D4 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19E1 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19E2 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| 19E4 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| 19F1 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 19F2 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| 19F4 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
| 38A1 | $19,2^{9},1$ | $228$ | $38$ | $27$ | $( 1, 3)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| 38A3 | $19,2^{9},1$ | $228$ | $38$ | $27$ | $( 1, 3)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 38A9 | $19,2^{9},1$ | $228$ | $38$ | $27$ | $( 1, 3)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 38B1 | $38$ | $228$ | $38$ | $37$ | $( 1,31, 7,32,13,33,19,34, 6,35,12,36,18,37, 5,38,11,20,17,21, 4,22,10,23,16,24, 3,25, 9,26,15,27, 2,28, 8,29,14,30)$ |
| 38B3 | $38$ | $228$ | $38$ | $37$ | $( 1,32,19,35,18,38,17,22,16,25,15,28,14,31,13,34,12,37,11,21,10,24, 9,27, 8,30, 7,33, 6,36, 5,20, 4,23, 3,26, 2,29)$ |
| 38B9 | $38$ | $228$ | $38$ | $37$ | $( 1,35,17,25,14,34,11,24, 8,33, 5,23, 2,32,18,22,15,31,12,21, 9,30, 6,20, 3,29,19,38,16,28,13,37,10,27, 7,36, 4,26)$ |
Malle's constant $a(G)$: $1/9$
magma: ConjugacyClasses(G);
Character table
39 x 39 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed