Show commands:
Magma
magma: G := TransitiveGroup(38, 23);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $D_{19}^2:S_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| 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);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $24$: $(C_6\times C_2):C_2$ 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 | Representative |
| $1^{38}$ | $1$ | $1$ | $()$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,38,37,36,35,34, 33,32,31,30,29,28,27,26,25,24,23,22,21)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ | |
| $19^{2}$ | $12$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ | |
| $19,1^{19}$ | $12$ | $19$ | $(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,38,37,36,35,34, 33,32,31,30,29,28,27,26,25,24,23,22,21)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ | |
| $19,1^{19}$ | $12$ | $19$ | $(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ | |
| $19,1^{19}$ | $12$ | $19$ | $(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ | |
| $19^{2}$ | $24$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ | |
| $3^{12},1^{2}$ | $722$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ | |
| $38$ | $228$ | $38$ | $( 1,22,15,21,10,20, 5,38,19,37,14,36, 9,35, 4,34,18,33,13,32, 8,31, 3,30,17, 29,12,28, 7,27, 2,26,16,25,11,24, 6,23)$ | |
| $38$ | $228$ | $38$ | $( 1,21, 5,37, 9,34,13,31,17,28, 2,25, 6,22,10,38,14,35,18,32, 3,29, 7,26,11, 23,15,20,19,36, 4,33, 8,30,12,27,16,24)$ | |
| $38$ | $228$ | $38$ | $( 1,20,14,34, 8,29, 2,24,15,38, 9,33, 3,28,16,23,10,37, 4,32,17,27,11,22, 5, 36,18,31,12,26, 6,21,19,35,13,30, 7,25)$ | |
| $38$ | $228$ | $38$ | $( 1,37,13,28, 6,38,18,29,11,20, 4,30,16,21, 9,31, 2,22,14,32, 7,23,19,33,12, 24, 5,34,17,25,10,35, 3,26,15,36, 8,27)$ | |
| $38$ | $228$ | $38$ | $( 1,33,11,35, 2,37,12,20, 3,22,13,24, 4,26,14,28, 5,30,15,32, 6,34,16,36, 7, 38,17,21, 8,23,18,25, 9,27,19,29,10,31)$ | |
| $38$ | $228$ | $38$ | $( 1,28,18,20,16,31,14,23,12,34,10,26, 8,37, 6,29, 4,21, 2,32,19,24,17,35,15, 27,13,38,11,30, 9,22, 7,33, 5,25, 3,36)$ | |
| $38$ | $228$ | $38$ | $( 1,34, 2,38, 3,23, 4,27, 5,31, 6,35, 7,20, 8,24, 9,28,10,32,11,36,12,21,13, 25,14,29,15,33,16,37,17,22,18,26,19,30)$ | |
| $2^{19}$ | $114$ | $2$ | $( 1,32)( 2,36)( 3,21)( 4,25)( 5,29)( 6,33)( 7,37)( 8,22)( 9,26)(10,30)(11,34) (12,38)(13,23)(14,27)(15,31)(16,35)(17,20)(18,24)(19,28)$ | |
| $38$ | $228$ | $38$ | $( 1,26,17,33,14,21,11,28, 8,35, 5,23, 2,30,18,37,15,25,12,32, 9,20, 6,27, 3, 34,19,22,16,29,13,36,10,24, 7,31, 4,38)$ | |
| $38$ | $228$ | $38$ | $( 1,35,12,22, 4,28,15,34, 7,21,18,27,10,33, 2,20,13,26, 5,32,16,38, 8,25,19, 31,11,37, 3,24,14,30, 6,36,17,23, 9,29)$ | |
| $6^{6},1^{2}$ | $722$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ | |
| $2^{18},1^{2}$ | $361$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ | |
| $2^{9},1^{20}$ | $38$ | $2$ | $(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ | |
| $19,2^{9},1$ | $228$ | $38$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,38)(21,37) (22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)$ | |
| $19,2^{9},1$ | $228$ | $38$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,37)(21,36) (22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)$ | |
| $19,2^{9},1$ | $228$ | $38$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,35)(21,34) (22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(36,38)$ | |
| $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,28,27,38,31,32 )(22,36,34,37,23,25)(24,33,29,35,26,30)$ | |
| $6^{3},3^{6},1^{2}$ | $722$ | $6$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,32,31,38,27,28 )(22,25,23,37,34,36)(24,30,26,35,29,33)$ | |
| $4^{9},2$ | $2166$ | $4$ | $( 1,22,14,36)( 2,26,13,32)( 3,30,12,28)( 4,34,11,24)( 5,38,10,20)( 6,23, 9,35) ( 7,27, 8,31)(15,21,19,37)(16,25,18,33)(17,29)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $8664=2^{3} \cdot 3 \cdot 19^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 8664.x | magma: IdentifyGroup(G);
| |
| Character table: | 42 x 42 character table |
magma: CharacterTable(G);