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Magma
magma: G := TransitiveGroup(38, 32);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{19}^2:(S_3\times C_9)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31), (1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $9$: $C_9$ $18$: $S_3\times C_3$, $C_{18}$ $54$: $C_9\times S_3$ 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 | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 19, 19 $ | $54$ | $19$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ | |
| $ 19, 19 $ | $54$ | $19$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ | |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $19$ | $(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ | |
| $ 19, 19 $ | $27$ | $19$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ | |
| $ 19, 19 $ | $54$ | $19$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ | |
| $ 19, 19 $ | $54$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ | |
| $ 19, 19 $ | $27$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ | |
| $ 19, 19 $ | $54$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$ | |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $3$ | $(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ | |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $342$ | $57$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,30,24)(21,37,35) (22,25,27)(23,32,38)(26,34,33)(28,29,36)$ | |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $342$ | $57$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,21,28)(22,35,31) (24,30,34)(25,37,26)(27,32,29)(33,36,38)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ | |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $342$ | $57$ | $( 1, 5,11)( 2,16,18)( 3, 8, 6)( 4,19,13)( 7,14,15)( 9,17,10)(20,30,21,31,22, 32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ | |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $342$ | $57$ | $( 1, 9, 2)( 3,12,16)( 5,15,11)( 6, 7,18)( 8,10,13)(14,19,17)(20,21,22,23,24, 25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ | |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $1083$ | $6$ | $( 1,26,18,34, 4,33)( 2,22, 6,25,15,27)( 3,37,13,35, 7,21)( 5,29, 8,36,10,28) ( 9,32,17,38,16,23)(11,24,12,20,19,30)(14,31)$ | |
| $ 38 $ | $513$ | $38$ | $( 1,24, 2,34, 3,25, 4,35, 5,26, 6,36, 7,27, 8,37, 9,28,10,38,11,29,12,20,13, 30,14,21,15,31,16,22,17,32,18,23,19,33)$ | |
| $ 38 $ | $513$ | $38$ | $( 1,26, 3,27, 5,28, 7,29, 9,30,11,31,13,32,15,33,17,34,19,35, 2,36, 4,37, 6, 38, 8,20,10,21,12,22,14,23,16,24,18,25)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $57$ | $2$ | $( 1,22)( 2,32)( 3,23)( 4,33)( 5,24)( 6,34)( 7,25)( 8,35)( 9,26)(10,36)(11,27) (12,37)(13,28)(14,38)(15,29)(16,20)(17,30)(18,21)(19,31)$ | |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $1083$ | $6$ | $( 1,29, 2,23,13,33)( 3,36, 5,24, 8,25)( 4,30,16,34,15,21)( 6,37,19,35,10,32) ( 7,31,11,26,17,28)( 9,38,14,27,12,20)(18,22)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ | |
| $ 18, 18, 2 $ | $1083$ | $18$ | $( 1,26,13,24,15,30, 9,31, 8,28,11,37, 2,29,10,34, 5,38)( 3,32, 7,25,14,27,12, 21,18,20,19,23,16,33, 6,22,17,36)( 4,35)$ | |
| $ 18, 18, 2 $ | $1083$ | $18$ | $( 1,24, 4,30,19,22,18,20,13,29, 7,36,15,33,17,37, 8,38)( 2,26, 9,21, 6,34,10, 23,11,25,16,35, 3,28,14,31,12,27)( 5,32)$ | |
| $ 18, 18, 2 $ | $1083$ | $18$ | $( 1,29, 5,28,16,30,13,26,19,34, 7,37,12,31, 2,24, 3,38)( 4,33,18,20, 9,27, 8, 32,10,22, 6,23,14,21,17,25,11,36)(15,35)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ | |
| $ 18, 18, 2 $ | $1083$ | $18$ | $( 1,26, 2,38, 6,29, 3,31,10,20,19,33,17,28, 9,27,15,23)( 4,24,14,30,16,35, 5, 36,18,21,13,37,12,25, 8,34,11,32)( 7,22)$ | |
| $ 18, 18, 2 $ | $1083$ | $18$ | $( 1,24,16,30,11,28,19,35,10,20,13,25,12,36, 6,26, 8,23)( 2,32, 3,21, 9,31, 7, 34,14,33,18,27, 4,29,15,22, 5,37)(17,38)$ | |
| $ 18, 18, 2 $ | $1083$ | $18$ | $( 1,29, 4,26,12,37, 8,22,10,20, 9,21,19,30,14,35, 7,23)( 2,28,13,36,17,32,15, 34,16,33, 6,24,11,38,18,31, 5,25)( 3,27)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $19494=2 \cdot 3^{3} \cdot 19^{2}$ | 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 | ||
| Label: | 19494.i | magma: IdentifyGroup(G);
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| Character table: | 42 x 42 character table |
magma: CharacterTable(G);