Show commands:
Magma
magma: G := TransitiveGroup(26, 45);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $D_{13}^2.C_{12}$ | ||
| 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: | (1,19,13,22,11,15,7,14,12,25,9,21,3,26,4,23,6,17,10,18,5,20,8,24)(2,16), (1,21,3,25,7,20,2,23,5,16,11,15,10,26,8,22,4,14,9,24,6,18,13,19)(12,17) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ $16$: $C_8:C_2$ $24$: 24T2 $48$: 24T16 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: 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 |
| $1^{26}$ | $1$ | $1$ | $()$ | |
| $13^{2}$ | $48$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $13,1^{13}$ | $24$ | $13$ | $(14,16,18,20,22,24,26,15,17,19,21,23,25)$ | |
| $13^{2}$ | $48$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $13^{2}$ | $48$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
| $3^{8},1^{2}$ | $169$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ | |
| $3^{8},1^{2}$ | $169$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ | |
| $6^{4},1^{2}$ | $169$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$ | |
| $2^{12},1^{2}$ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $6^{4},1^{2}$ | $169$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $12^{2},1^{2}$ | $169$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,16,18,22,17,20,26,25,23,19,24,21)$ | |
| $12^{2},1^{2}$ | $169$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,20,24,22,23,16,26,21,17,19,18,25)$ | |
| $4^{6},1^{2}$ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ | |
| $4^{6},1^{2}$ | $169$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ | |
| $12^{2},1^{2}$ | $169$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,25,18,19,17,21,26,16,23,22,24,20)$ | |
| $12^{2},1^{2}$ | $169$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,21,24,19,23,25,26,20,17,22,18,16)$ | |
| $24,2$ | $338$ | $24$ | $( 1,19,13,22,11,15, 7,14,12,25, 9,21, 3,26, 4,23, 6,17,10,18, 5,20, 8,24) ( 2,16)$ | |
| $8^{3},2$ | $338$ | $8$ | $( 1,16, 4,15, 6,23, 3,24)( 2,20, 9,22, 5,19,11,17)( 7,14, 8,18,13,25,12,21) (10,26)$ | |
| $24,2$ | $338$ | $24$ | $( 1,20,12,22,13,21, 6,15, 3,18,11,23, 7,14, 9,25, 8,26, 2,19, 5,16,10,24) ( 4,17)$ | |
| $8^{3},2$ | $338$ | $8$ | $( 1,24)( 2,18, 9,15,13,17, 6,20)( 3,25, 4,19,12,23,11,16)( 5,26, 7,14,10,22, 8,21)$ | |
| $24,2$ | $338$ | $24$ | $( 1,18,12,15,11,20, 4,16, 7,14, 2,26, 6,19, 8,22, 9,17, 3,21,13,23, 5,24) (10,25)$ | |
| $24,2$ | $338$ | $24$ | $( 1,26, 3,22,12,17, 7,14, 4,20,10,21,11,19, 9,23,13,15, 5,18, 8,25, 2,24) ( 6,16)$ | |
| $2^{6},1^{14}$ | $26$ | $2$ | $(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ | |
| $13,2^{6},1$ | $312$ | $26$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25)(15,24)(16,23)(17,22)(18,21) (19,20)$ | |
| $6^{2},3^{4},1^{2}$ | $338$ | $6$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $6^{2},3^{4},1^{2}$ | $338$ | $6$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ | |
| $12^{2},1^{2}$ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,25,18,19,17,21,26,16,23,22,24,20)$ | |
| $12^{2},1^{2}$ | $338$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,21,24,19,23,25,26,20,17,22,18,16)$ | |
| $4^{6},1^{2}$ | $338$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ | |
| $24,2$ | $338$ | $24$ | $( 1,19,11,15, 4,23, 5,20, 3,26, 7,14,12,25, 2,16, 9,21, 8,24,10,18, 6,17) (13,22)$ | |
| $8^{3},2$ | $338$ | $8$ | $( 1,16,12,21, 9,22,11,17)( 2,20, 7,14, 8,18, 3,24)( 4,15,10,26, 6,23,13,25) ( 5,19)$ | |
| $24,2$ | $338$ | $24$ | $( 1,20, 6,15, 2,19,13,21,12,22, 5,16, 8,26, 3,18, 7,14, 9,25,10,24, 4,17) (11,23)$ | |
| $8^{3},2$ | $338$ | $8$ | $( 1,24, 6,20, 5,26,13,17)( 2,18,11,16, 4,19, 8,21)( 3,25)( 7,14,10,22,12,23, 9,15)$ | |
| $24,2$ | $338$ | $24$ | $( 1,18, 5,24, 3,21, 4,16,10,25, 7,14, 2,26,11,20,13,23,12,15, 6,19, 9,17) ( 8,22)$ | |
| $24,2$ | $338$ | $24$ | $( 1,26, 5,18,13,15, 3,22, 9,23, 8,25, 6,16, 2,24, 7,14, 4,20,11,19,12,17) (10,21)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $8112=2^{4} \cdot 3 \cdot 13^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 8112.bf | magma: IdentifyGroup(G);
| |
| Character table: | 35 x 35 character table |
magma: CharacterTable(G);