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Magma
magma: G := TransitiveGroup(38, 23);
Group invariants
| Abstract group: | $D_{19}^2:S_3$ | 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 $t$: | $23$ | 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: | $(1,22,15,21,10,20,5,38,19,37,14,36,9,35,4,34,18,33,13,32,8,31,3,30,17,29,12,28,7,27,2,26,16,25,11,24,6,23)$, $(1,11,15,9,18,14)(2,19,3,8,10,7)(4,16,17,6,13,12)(20,29,35)(21,36,27)(22,24,38)(23,31,30)(25,26,33)(32,37,34)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $24$: $(C_6\times C_2):C_2$ 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
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{38}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{9},1^{20}$ | $38$ | $2$ | $9$ | $( 1, 2)( 3,19)( 4,18)( 5,17)( 6,16)( 7,15)( 8,14)( 9,13)(10,12)$ |
| 2B | $2^{19}$ | $114$ | $2$ | $19$ | $( 1,28)( 2,22)( 3,35)( 4,29)( 5,23)( 6,36)( 7,30)( 8,24)( 9,37)(10,31)(11,25)(12,38)(13,32)(14,26)(15,20)(16,33)(17,27)(18,21)(19,34)$ |
| 2C | $2^{18},1^{2}$ | $361$ | $2$ | $18$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,19)(10,18)(11,17)(12,16)(13,15)(20,21)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)$ |
| 3A | $3^{12},1^{2}$ | $722$ | $3$ | $24$ | $( 1,15,17)( 2, 7, 5)( 3,18,12)( 4,10,19)( 6,13,14)( 8,16, 9)(20,24,33)(21,31,25)(22,38,36)(23,26,28)(27,35,34)(29,30,37)$ |
| 4A | $4^{9},2$ | $2166$ | $4$ | $28$ | $( 1,35, 8,25)( 2,20, 7,21)( 3,24, 6,36)( 4,28, 5,32)( 9,29,19,31)(10,33,18,27)(11,37,17,23)(12,22,16,38)(13,26,15,34)(14,30)$ |
| 6A | $6^{6},1^{2}$ | $722$ | $6$ | $30$ | $( 1, 3,19,14,12,15)( 2,11, 7,13, 4, 8)( 5,16, 9,10,18, 6)(20,35,25,38,23,33)(21,28,36,37,30,22)(24,26,31,34,32,27)$ |
| 6B1 | $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $27$ | $( 1, 5,15, 2,17, 7)( 3,10,18,19,12, 4)( 6, 8,13,16,14, 9)(20,33,24)(21,25,31)(22,36,38)(23,28,26)(27,34,35)(29,37,30)$ |
| 6B-1 | $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $27$ | $( 1, 7,17, 2,15, 5)( 3, 4,12,19,18,10)( 6, 9,14,16,13, 8)(20,24,33)(21,31,25)(22,38,36)(23,26,28)(27,35,34)(29,30,37)$ |
| 19A1 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| 19A2 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$ |
| 19A4 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| 19B1 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| 19B2 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| 19B3 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| 19B4 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| 19B5 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 19B6 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| 19B7 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| 19B8 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19B9 | $19^{2}$ | $12$ | $19$ | $36$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| 19C1 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19C2 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| 19C3 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 19C4 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 19C5 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 19C6 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| 19C7 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$ |
| 19C8 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$ |
| 19C9 | $19^{2}$ | $24$ | $19$ | $36$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| 38A1 | $19,2^{9},1$ | $228$ | $38$ | $27$ | $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,19)(11,18)(12,17)(13,16)(14,15)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 38A3 | $19,2^{9},1$ | $228$ | $38$ | $27$ | $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,19)(11,18)(12,17)(13,16)(14,15)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 38A9 | $19,2^{9},1$ | $228$ | $38$ | $27$ | $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,19)(11,18)(12,17)(13,16)(14,15)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| 38B1 | $38$ | $228$ | $38$ | $37$ | $( 1,36,11,33, 2,30,12,27, 3,24,13,21, 4,37,14,34, 5,31,15,28, 6,25,16,22, 7,38,17,35, 8,32,18,29, 9,26,19,23,10,20)$ |
| 38B3 | $38$ | $228$ | $38$ | $37$ | $( 1,33,12,24, 4,34,15,25, 7,35,18,26,10,36, 2,27,13,37, 5,28,16,38, 8,29,19,20,11,30, 3,21,14,31, 6,22,17,32, 9,23)$ |
| 38B5 | $38$ | $228$ | $38$ | $37$ | $( 1,30,13,34, 6,38,18,23,11,27, 4,31,16,35, 9,20, 2,24,14,28, 7,32,19,36,12,21, 5,25,17,29,10,33, 3,37,15,22, 8,26)$ |
| 38B7 | $38$ | $228$ | $38$ | $37$ | $( 1,27,14,25, 8,23, 2,21,15,38, 9,36, 3,34,16,32,10,30, 4,28,17,26,11,24, 5,22,18,20,12,37, 6,35,19,33,13,31, 7,29)$ |
| 38B9 | $38$ | $228$ | $38$ | $37$ | $( 1,24,15,35,10,27, 5,38,19,30,14,22, 9,33, 4,25,18,36,13,28, 8,20, 3,31,17,23,12,34, 7,26, 2,37,16,29,11,21, 6,32)$ |
| 38B11 | $38$ | $228$ | $38$ | $37$ | $( 1,21,16,26,12,31, 8,36, 4,22,19,27,15,32,11,37, 7,23, 3,28,18,33,14,38,10,24, 6,29, 2,34,17,20,13,25, 9,30, 5,35)$ |
| 38B13 | $38$ | $228$ | $38$ | $37$ | $( 1,37,17,36,14,35,11,34, 8,33, 5,32, 2,31,18,30,15,29,12,28, 9,27, 6,26, 3,25,19,24,16,23,13,22,10,21, 7,20, 4,38)$ |
| 38B15 | $38$ | $228$ | $38$ | $37$ | $( 1,34,18,27,16,20,14,32,12,25,10,37, 8,30, 6,23, 4,35, 2,28,19,21,17,33,15,26,13,38,11,31, 9,24, 7,36, 5,29, 3,22)$ |
| 38B17 | $38$ | $228$ | $38$ | $37$ | $( 1,31,19,37,18,24,17,30,16,36,15,23,14,29,13,35,12,22,11,28,10,34, 9,21, 8,27, 7,33, 6,20, 5,26, 4,32, 3,38, 2,25)$ |
Malle's constant $a(G)$: $1/9$
magma: ConjugacyClasses(G);
Character table
42 x 42 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed