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Magma
magma: G := TransitiveGroup(38, 9);
Group invariants
| Abstract group: | $C_2\times F_{19}$ | magma: IdentifyGroup(G);
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| 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 | 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 $t$: | $9$ | 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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| $\card{\Aut(F/K)}$: | $2$ | 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
$\card{(G/N)}$ 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 $342$: $F_{19}$ 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
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{38}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{19}$ | $1$ | $2$ | $19$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$ |
| 2B | $2^{19}$ | $19$ | $2$ | $19$ | $( 1, 5)( 2, 6)( 3, 4)( 7,38)( 8,37)( 9,35)(10,36)(11,33)(12,34)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,25)(20,26)(21,24)(22,23)$ |
| 2C | $2^{18},1^{2}$ | $19$ | $2$ | $18$ | $( 3,38)( 4,37)( 5,36)( 6,35)( 7,34)( 8,33)( 9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$ |
| 3A1 | $3^{12},1^{2}$ | $19$ | $3$ | $24$ | $( 3,23,16)( 4,24,15)( 5, 7,29)( 6, 8,30)( 9,14,20)(10,13,19)(11,36,34)(12,35,33)(17,25,37)(18,26,38)(21,31,28)(22,32,27)$ |
| 3A-1 | $3^{12},1^{2}$ | $19$ | $3$ | $24$ | $( 3,16,23)( 4,15,24)( 5,29, 7)( 6,30, 8)( 9,20,14)(10,19,13)(11,34,36)(12,33,35)(17,37,25)(18,38,26)(21,28,31)(22,27,32)$ |
| 6A1 | $6^{6},2$ | $19$ | $6$ | $31$ | $( 1,29,23,27,38, 5)( 2,30,24,28,37, 6)( 3,15, 8,25,13,22)( 4,16, 7,26,14,21)( 9,12,36,19,17,31)(10,11,35,20,18,32)(33,34)$ |
| 6A-1 | $6^{6},2$ | $19$ | $6$ | $31$ | $( 1, 7,18,22,16, 5)( 2, 8,17,21,15, 6)( 3,24,31,20,38,29)( 4,23,32,19,37,30)( 9,33,36,13,27,26)(10,34,35,14,28,25)(11,12)$ |
| 6B1 | $6^{6},1^{2}$ | $19$ | $6$ | $30$ | $( 3,18,16,38,23,26)( 4,17,15,37,24,25)( 5,34,29,36, 7,11)( 6,33,30,35, 8,12)( 9,27,20,32,14,22)(10,28,19,31,13,21)$ |
| 6B-1 | $6^{6},1^{2}$ | $19$ | $6$ | $30$ | $( 3,26,23,38,16,18)( 4,25,24,37,15,17)( 5,11, 7,36,29,34)( 6,12, 8,35,30,33)( 9,22,14,32,20,27)(10,21,13,31,19,28)$ |
| 6C1 | $6^{6},2$ | $19$ | $6$ | $31$ | $( 1,11, 6, 2,12, 5)( 3,25,28, 4,26,27)( 7,16,34, 8,15,33)( 9,30,17,10,29,18)(13,20,23,14,19,24)(21,37,35,22,38,36)(31,32)$ |
| 6C-1 | $6^{6},2$ | $19$ | $6$ | $31$ | $( 1,34, 6, 2,33, 5)( 3,17,19, 4,18,20)( 7,23, 9, 8,24,10)(11,30,37,12,29,38)(13,14)(15,35,27,16,36,28)(21,25,31,22,26,32)$ |
| 9A1 | $9^{4},1^{2}$ | $19$ | $9$ | $32$ | $( 3,33,19,23,12,10,16,35,13)( 4,34,20,24,11, 9,15,36,14)( 5,27,37, 7,22,17,29,32,25)( 6,28,38, 8,21,18,30,31,26)$ |
| 9A-1 | $9^{4},1^{2}$ | $19$ | $9$ | $32$ | $( 3,10,33,16,19,35,23,13,12)( 4, 9,34,15,20,36,24,14,11)( 5,17,27,29,37,32, 7,25,22)( 6,18,28,30,38,31, 8,26,21)$ |
| 9A2 | $9^{4},1^{2}$ | $19$ | $9$ | $32$ | $( 3,19,12,16,13,33,23,10,35)( 4,20,11,15,14,34,24, 9,36)( 5,37,22,29,25,27, 7,17,32)( 6,38,21,30,26,28, 8,18,31)$ |
| 9A-2 | $9^{4},1^{2}$ | $19$ | $9$ | $32$ | $( 3,35,10,23,33,13,16,12,19)( 4,36, 9,24,34,14,15,11,20)( 5,32,17, 7,27,25,29,22,37)( 6,31,18, 8,28,26,30,21,38)$ |
| 9A4 | $9^{4},1^{2}$ | $19$ | $9$ | $32$ | $( 3,13,35,16,10,12,23,19,33)( 4,14,36,15, 9,11,24,20,34)( 5,25,32,29,17,22, 7,37,27)( 6,26,31,30,18,21, 8,38,28)$ |
| 9A-4 | $9^{4},1^{2}$ | $19$ | $9$ | $32$ | $( 3,12,13,23,35,19,16,33,10)( 4,11,14,24,36,20,15,34, 9)( 5,22,25, 7,32,37,29,27,17)( 6,21,26, 8,31,38,30,28,18)$ |
| 18A1 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,17,30,20,21,14, 8,32,12,15,38,25,35,34, 3, 9,23, 5)( 2,18,29,19,22,13, 7,31,11,16,37,26,36,33, 4,10,24, 6)(27,28)$ |
| 18A-1 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,22,35, 4,12, 9,19, 7,30,34,13,37,31,24,26,15,28, 5)( 2,21,36, 3,11,10,20, 8,29,33,14,38,32,23,25,16,27, 6)(17,18)$ |
| 18A5 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1, 9,31,25,38,14,23, 4, 6, 2,10,32,26,37,13,24, 3, 5)( 7,35,17,16,20,12,27,33,22, 8,36,18,15,19,11,28,34,21)(29,30)$ |
| 18A-5 | $18^{2},1^{2}$ | $19$ | $18$ | $34$ | $( 3,21,12,26,13, 8,23,31,35,38,19,30,16,28,33,18,10, 6)( 4,22,11,25,14, 7,24,32,36,37,20,29,15,27,34,17, 9, 5)$ |
| 18A7 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,24,35, 7,10,17,12,25, 6, 2,23,36, 8, 9,18,11,26, 5)( 3,32,30,22,28,14,33,37,16, 4,31,29,21,27,13,34,38,15)(19,20)$ |
| 18A-7 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,25,33,24, 8,27,21,20,31,36,12, 4,13,29,10,15,18, 5)( 2,26,34,23, 7,28,22,19,32,35,11, 3,14,30, 9,16,17, 6)(37,38)$ |
| 18B1 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,27,30,36,16,32, 3,34,10,14,26,24,18,37,21,11,19, 5)( 2,28,29,35,15,31, 4,33, 9,13,25,23,17,38,22,12,20, 6)( 7, 8)$ |
| 18B-1 | $18^{2},1^{2}$ | $19$ | $18$ | $34$ | $( 3,31,33,26,19, 6,23,28,12,38,10, 8,16,21,35,18,13,30)( 4,32,34,25,20, 5,24,27,11,37, 9, 7,15,22,36,17,14,29)$ |
| 18B5 | $18^{2},1^{2}$ | $19$ | $18$ | $34$ | $( 3, 6,10,18,33,28,16,30,19,38,35,31,23, 8,13,26,12,21)( 4, 5, 9,17,34,27,15,29,20,37,36,32,24, 7,14,25,11,22)$ |
| 18B-5 | $18^{2},1^{2}$ | $19$ | $18$ | $34$ | $( 3, 8,19,18,12,31,16, 6,13,38,33,21,23,30,10,26,35,28)( 4, 7,20,17,11,32,15, 5,14,37,34,22,24,29, 9,25,36,27)$ |
| 18B7 | $18^{2},1^{2}$ | $19$ | $18$ | $34$ | $( 3,30,13,18,35,21,16, 8,10,38,12,28,23, 6,19,26,33,31)( 4,29,14,17,36,22,15, 7, 9,37,11,27,24, 5,20,25,34,32)$ |
| 18B-7 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,32,16,22,33,20,30,11,13,17,26, 4,35,24,38,27, 8, 5)( 2,31,15,21,34,19,29,12,14,18,25, 3,36,23,37,28, 7, 6)( 9,10)$ |
| 18C1 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,14,16, 9,28,11,21,29, 6, 2,13,15,10,27,12,22,30, 5)( 3, 7,33,32,38,20,35,25,18, 4, 8,34,31,37,19,36,26,17)(23,24)$ |
| 18C-1 | $18^{2},1^{2}$ | $19$ | $18$ | $34$ | $( 3,28,35,26,10,30,23,21,33,38,13, 6,16,31,12,18,19, 8)( 4,27,36,25, 9,29,24,22,34,37,14, 5,15,32,11,17,20, 7)$ |
| 18C5 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,15,23,34,18,36,30,32, 6, 2,16,24,33,17,35,29,31, 5)( 3,27,19, 9,26, 7,13,11,38, 4,28,20,10,25, 8,14,12,37)(21,22)$ |
| 18C-5 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,37,18, 7,21, 9, 3,20,28,32,33,15,26,11,23,29,13, 5)( 2,38,17, 8,22,10, 4,19,27,31,34,16,25,12,24,30,14, 6)(35,36)$ |
| 18C7 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1, 4,21,32, 8,20,13,36, 6, 2, 3,22,31, 7,19,14,35, 5)( 9,38,24,12,17,33,25,30,27,10,37,23,11,18,34,26,29,28)(15,16)$ |
| 18C-7 | $18^{2},2$ | $19$ | $18$ | $35$ | $( 1,20,33,27,35,37,10,22, 6, 2,19,34,28,36,38, 9,21, 5)( 3,29, 8,11,31,17,23,15,13, 4,30, 7,12,32,18,24,16,14)(25,26)$ |
| 19A | $19^{2}$ | $18$ | $19$ | $36$ | $( 1,31,23,16, 8,38,30,21,13, 6,35,28,19,12, 3,33,26,18,10)( 2,32,24,15, 7,37,29,22,14, 5,36,27,20,11, 4,34,25,17, 9)$ |
| 38A | $38$ | $18$ | $38$ | $37$ | $( 1,36,31,27,23,20,16,11, 8, 4,38,34,30,25,21,17,13, 9, 6, 2,35,32,28,24,19,15,12, 7, 3,37,33,29,26,22,18,14,10, 5)$ |
Malle's constant $a(G)$: $1/18$
magma: ConjugacyClasses(G);
Character table
38 x 38 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed