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Magma
magma: G := TransitiveGroup(38, 34);
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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| Group: | $D_{19}^2:D_9$ | ||
| 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: | (1,4,3,16,18,11,7,2,10)(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
|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 | 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 $ | $36$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ | |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ | |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $19$ | $(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ | |
| $ 9, 9, 9, 9, 1, 1 $ | $722$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $2$ | $(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ | |
| $ 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $684$ | $38$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,25)(21,24) (22,23)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $6$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,32,31,38,27,28 )(22,25,23,37,34,36)(24,30,26,35,29,33)$ | |
| $ 6, 6, 6, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $6$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,28,27,38,31,32 )(22,36,34,37,23,25)(24,33,29,35,26,30)$ | |
| $ 18, 9, 9, 1, 1 $ | $722$ | $18$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,30,25,32,26,23,31, 35,37,38,29,34,27,33,36,28,24,22)$ | |
| $ 18, 9, 9, 1, 1 $ | $722$ | $18$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,33,37,32,24,34,31, 30,36,38,26,22,27,35,25,28,29,23)$ | |
| $ 18, 9, 9, 1, 1 $ | $722$ | $18$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,35,36,32,29,22,31, 33,25,38,24,23,27,30,37,28,26,34)$ | |
| $ 18, 9, 9, 1, 1 $ | $722$ | $18$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$ | |
| $ 18, 9, 9, 1, 1 $ | $722$ | $18$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$ | |
| $ 18, 9, 9, 1, 1 $ | $722$ | $18$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$ | |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)(21,33,37,32,24,34,31, 30,36,38,26,22,27,35,25,28,29,23)$ | |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)(21,35,36,32,29,22,31, 33,25,38,24,23,27,30,37,28,26,34)$ | |
| $ 18, 18, 1, 1 $ | $722$ | $18$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(21,30,25,32,26,23,31, 35,37,38,29,34,27,33,36,28,24,22)$ | |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $722$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $361$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,34,17,33,14,32,11,31, 8,30, 5,29, 2,28,18,27,15,26,12,25, 9,24, 6,23, 3, 22,19,21,16,20,13,38,10,37, 7,36, 4,35)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,20, 7,22,13,24,19,26, 6,28,12,30,18,32, 5,34,11,36,17,38, 4,21,10,23,16, 25, 3,27, 9,29,15,31, 2,33, 8,35,14,37)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,25,16,30,12,35, 8,21, 4,26,19,31,15,36,11,22, 7,27, 3,32,18,37,14,23,10, 28, 6,33, 2,38,17,24,13,29, 9,34, 5,20)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,35,15,27,10,38, 5,30,19,22,14,33, 9,25, 4,36,18,28,13,20, 8,31, 3,23,17, 34,12,26, 7,37, 2,29,16,21,11,32, 6,24)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,36,13,21, 6,25,18,29,11,33, 4,37,16,22, 9,26, 2,30,14,34, 7,38,19,23,12, 27, 5,31,17,35,10,20, 3,24,15,28, 8,32)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,38, 9,28,17,37, 6,27,14,36, 3,26,11,35,19,25, 8,34,16,24, 5,33,13,23, 2, 32,10,22,18,31, 7,21,15,30, 4,20,12,29)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $342$ | $2$ | $( 1,23)( 2,36)( 3,30)( 4,24)( 5,37)( 6,31)( 7,25)( 8,38)( 9,32)(10,26)(11,20) (12,33)(13,27)(14,21)(15,34)(16,28)(17,22)(18,35)(19,29)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,28,10,31,19,34, 9,37,18,21, 8,24,17,27, 7,30,16,33, 6,36,15,20, 5,23,14, 26, 4,29,13,32, 3,35,12,38, 2,22,11,25)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,22, 3,29, 5,36, 7,24, 9,31,11,38,13,26,15,33,17,21,19,28, 2,35, 4,23, 6, 30, 8,37,10,25,12,32,14,20,16,27,18,34)$ | |
| $ 38 $ | $684$ | $38$ | $( 1,33,19,20,18,26,17,32,16,38,15,25,14,31,13,37,12,24,11,30,10,36, 9,23, 8, 29, 7,35, 6,22, 5,28, 4,34, 3,21, 2,27)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $6498$ | $4$ | $( 1,25, 9,35)( 2,31, 8,29)( 3,37, 7,23)( 4,24, 6,36)( 5,30)(10,22,19,38) (11,28,18,32)(12,34,17,26)(13,21,16,20)(14,27,15,33)$ |
magma: ConjugacyClasses(G);
Group invariants
| 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 | ||
| Label: | 25992.bh | magma: IdentifyGroup(G);
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| Character table: | 41 x 41 character table |
magma: CharacterTable(G);