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Magma
magma: G := TransitiveGroup(38, 34);
Group invariants
| Abstract group: | $D_{19}^2:D_9$ | magma: IdentifyGroup(G);
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| Order: | $25992=2^{3} \cdot 3^{2} \cdot 19^{2}$ | 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 $t$: | $34$ | 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: | $(1,4,3,16,18,11,7,2,10)(5,9,14,6,15,12,13,19,17)(20,22,28,27,24,34,26,21,25,37,35,29,30,33,23,31,36,32)$, $(1,34,17,33,14,32,11,31,8,30,5,29,2,28,18,27,15,26,12,25,9,24,6,23,3,22,19,21,16,20,13,38,10,37,7,36,4,35)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $18$: $D_{9}$ $24$: $(C_6\times C_2):C_2$ $36$: $D_{18}$ $72$: 36T24 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 | $2^{9},1^{20}$ | $38$ | $2$ | $9$ | $(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| 2B | $2^{19}$ | $342$ | $2$ | $19$ | $( 1,35)( 2,29)( 3,23)( 4,36)( 5,30)( 6,24)( 7,37)( 8,31)( 9,25)(10,38)(11,32)(12,26)(13,20)(14,33)(15,27)(16,21)(17,34)(18,28)(19,22)$ |
| 2C | $2^{18},1^{2}$ | $361$ | $2$ | $18$ | $( 1, 6)( 2, 5)( 3, 4)( 7,19)( 8,18)( 9,17)(10,16)(11,15)(12,14)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| 3A | $3^{12},1^{2}$ | $722$ | $3$ | $24$ | $( 1,19, 8)( 2,11,15)( 4,14,10)( 5, 6,17)( 7, 9,12)(13,18,16)(20,21,28)(22,35,31)(24,30,34)(25,37,26)(27,32,29)(33,36,38)$ |
| 4A | $4^{9},2$ | $6498$ | $4$ | $28$ | $( 1,24, 6,35)( 2,30, 5,29)( 3,36, 4,23)( 7,22,19,37)( 8,28,18,31)( 9,34,17,25)(10,21,16,38)(11,27,15,32)(12,33,14,26)(13,20)$ |
| 6A | $6^{6},1^{2}$ | $722$ | $6$ | $30$ | $( 1, 7,17, 2,15, 5)( 3, 4,12,19,18,10)( 6, 9,14,16,13, 8)(20,38,26,34,35,28)(21,31,37,33,23,36)(22,24,29,32,30,25)$ |
| 6B1 | $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $27$ | $( 1, 8,19)( 2,15,11)( 4,10,14)( 5,17, 6)( 7,12, 9)(13,16,18)(20,31,24,25,33,21)(22,28,38,23,36,26)(27,30,35,37,34,29)$ |
| 6B-1 | $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $27$ | $( 1,19, 8)( 2,11,15)( 4,14,10)( 5, 6,17)( 7, 9,12)(13,18,16)(20,38,26,34,35,28)(21,31,37,33,23,36)(22,24,29,32,30,25)$ |
| 9A1 | $9^{4},1^{2}$ | $722$ | $9$ | $32$ | $( 1,12,10,19, 7, 4, 8, 9,14)( 2,17,16,11, 5,13,15, 6,18)(20,30,32,21,34,29,28,24,27)(22,38,26,35,33,25,31,36,37)$ |
| 9A2 | $9^{4},1^{2}$ | $722$ | $9$ | $32$ | $( 1, 9, 4,19,12,14, 8, 7,10)( 2, 6,13,11,17,18,15, 5,16)(20,24,29,21,30,27,28,34,32)(22,36,25,35,38,37,31,33,26)$ |
| 9A4 | $9^{4},1^{2}$ | $722$ | $9$ | $32$ | $( 1, 7,14,19, 9,10, 8,12, 4)( 2, 5,18,11, 6,16,15,17,13)(20,34,27,21,24,32,28,30,29)(22,33,37,35,36,26,31,38,25)$ |
| 18A1 | $18,9^{2},1^{2}$ | $722$ | $18$ | $33$ | $( 1, 9, 4,19,12,14, 8, 7,10)( 2, 6,13,11,17,18,15, 5,16)(20,35,21,29,38,22,23,36,34,27,31,26,37,28,25,24,30,32)$ |
| 18A5 | $18^{2},1^{2}$ | $722$ | $18$ | $34$ | $( 1,17, 8,19,14,18,11, 9, 3, 4, 7,16, 5,10, 6,13,15, 2)(20,35,21,29,38,22,23,36,34,27,31,26,37,28,25,24,30,32)$ |
| 18A7 | $18,9^{2},1^{2}$ | $722$ | $18$ | $33$ | $( 1, 4,12, 8,10, 9,19,14, 7)( 2,13,17,15,16, 6,11,18, 5)(20,30,31,33,37,26,23,36,24,38,28,27,25,21,32,35,22,34)$ |
| 18B1 | $18^{2},1^{2}$ | $722$ | $18$ | $34$ | $( 1,19,17,13, 5, 8,14, 7,12, 3, 4, 6,10,18,15, 9,16,11)(20,25,37,24,27,38,34,32,31,21,35,23,36,33,22,26,28,29)$ |
| 18B-1 | $18,9^{2},1^{2}$ | $722$ | $18$ | $33$ | $( 1, 7,14,19, 9,10, 8,12, 4)( 2, 5,18,11, 6,16,15,17,13)(20,25,37,24,27,38,34,32,31,21,35,23,36,33,22,26,28,29)$ |
| 18B5 | $18,9^{2},1^{2}$ | $722$ | $18$ | $33$ | $( 1,12,10,19, 7, 4, 8, 9,14)( 2,17,16,11, 5,13,15, 6,18)(20,29,31,23,36,22,21,25,28,35,26,24,32,38,33,34,30,27)$ |
| 18B-5 | $18,9^{2},1^{2}$ | $722$ | $18$ | $33$ | $( 1,10, 7, 8,14,12,19, 4, 9)( 2,16, 5,15,18,17,11,13, 6)(20,27,29,35,34,31,22,33,28,32,25,23,36,37,21,30,38,24)$ |
| 18B7 | $18,9^{2},1^{2}$ | $722$ | $18$ | $33$ | $( 1,14, 9, 8, 4, 7,19,10,12)( 2,18, 6,15,13, 5,11,16,17)(20,32,29,25,26,21,27,35,33,24,31,34,38,37,23,36,28,30)$ |
| 18B-7 | $18^{2},1^{2}$ | $722$ | $18$ | $34$ | $( 1,14, 6, 8,17,10, 7, 3, 4,18, 5,13,11, 2, 9,12,16,15)(20,29,31,23,36,22,21,25,28,35,26,24,32,38,33,34,30,27)$ |
| 19A | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| 19B1 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| 19B2 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| 19B3 | $19,1^{19}$ | $36$ | $19$ | $18$ | $(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| 19B4 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 19B5 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19B6 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| 19B7 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| 19B8 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| 19B9 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 38A | $38$ | $684$ | $38$ | $37$ | $( 1,35, 5,30, 9,25,13,20,17,34, 2,29, 6,24,10,38,14,33,18,28, 3,23, 7,37,11,32,15,27,19,22, 4,36, 8,31,12,26,16,21)$ |
| 38B1 | $38$ | $684$ | $38$ | $37$ | $( 1,35,19,22,18,28,17,34,16,21,15,27,14,33,13,20,12,26,11,32,10,38, 9,25, 8,31, 7,37, 6,24, 5,30, 4,36, 3,23, 2,29)$ |
| 38B3 | $38$ | $684$ | $38$ | $37$ | $( 1,35, 6,24,11,32,16,21, 2,29, 7,37,12,26,17,34, 3,23, 8,31,13,20,18,28, 4,36, 9,25,14,33,19,22, 5,30,10,38,15,27)$ |
| 38B5 | $38$ | $684$ | $38$ | $37$ | $( 1,35,11,32, 2,29,12,26, 3,23,13,20, 4,36,14,33, 5,30,15,27, 6,24,16,21, 7,37,17,34, 8,31,18,28, 9,25,19,22,10,38)$ |
| 38B7 | $19,2^{9},1$ | $684$ | $38$ | $27$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,29)(21,28)(22,27)(23,26)(24,25)(30,38)(31,37)(32,36)(33,35)$ |
| 38B9 | $38$ | $684$ | $38$ | $37$ | $( 1,35,18,28,16,21,14,33,12,26,10,38, 8,31, 6,24, 4,36, 2,29,19,22,17,34,15,27,13,20,11,32, 9,25, 7,37, 5,30, 3,23)$ |
| 38B11 | $38$ | $684$ | $38$ | $37$ | $( 1,35,13,20, 6,24,18,28,11,32, 4,36,16,21, 9,25, 2,29,14,33, 7,37,19,22,12,26, 5,30,17,34,10,38, 3,23,15,27, 8,31)$ |
| 38B13 | $38$ | $684$ | $38$ | $37$ | $( 1,35,12,26, 4,36,15,27, 7,37,18,28,10,38, 2,29,13,20, 5,30,16,21, 8,31,19,22,11,32, 3,23,14,33, 6,24,17,34, 9,25)$ |
| 38B15 | $38$ | $684$ | $38$ | $37$ | $( 1,35,17,34,14,33,11,32, 8,31, 5,30, 2,29,18,28,15,27,12,26, 9,25, 6,24, 3,23,19,22,16,21,13,20,10,38, 7,37, 4,36)$ |
| 38B17 | $38$ | $684$ | $38$ | $37$ | $( 1,35, 7,37,13,20,19,22, 6,24,12,26,18,28, 5,30,11,32,17,34, 4,36,10,38,16,21, 3,23, 9,25,15,27, 2,29, 8,31,14,33)$ |
Malle's constant $a(G)$: $1/9$
magma: ConjugacyClasses(G);
Character table
41 x 41 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed