Show commands:
Magma
magma: G := TransitiveGroup(26, 29);
Group invariants
| Abstract group: | $C_{13}^2:C_{24}$ | magma: IdentifyGroup(G);
| |
| Order: | $4056=2^{3} \cdot 3 \cdot 13^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | magma: NilpotencyClass(G);
|
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| |
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | $(1,2)(3,13)(4,12)(5,11)(6,10)(7,9)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)$, $(1,26,11,21,3,25,12,14,10,15,9,22,2,19,5,24,13,20,4,18,6,17,7,23)(8,16)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $8$: $C_8$ $12$: $C_{12}$ $24$: $C_{24}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
26T29 x 6Siblings 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 | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 2)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)$ |
| 3A1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 1,10,13)( 2, 6, 3)( 4,11, 9)( 5, 7,12)(14,24,23)(15,20,26)(17,25,19)(18,21,22)$ |
| 3A-1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 1,13,10)( 2, 3, 6)( 4, 9,11)( 5,12, 7)(14,23,24)(15,26,20)(17,19,25)(18,22,21)$ |
| 4A1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 4, 2,12)( 3, 7,13, 9)( 5,10,11, 6)(14,26,18,19)(15,21,17,24)(20,22,25,23)$ |
| 4A-1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1,12, 2, 4)( 3, 9,13, 7)( 5, 6,11,10)(14,19,18,26)(15,24,17,21)(20,23,25,22)$ |
| 6A1 | $6^{4},1^{2}$ | $169$ | $6$ | $20$ | $( 1, 3,10, 2,13, 6)( 4, 7,11,12, 9, 5)(14,22,24,18,23,21)(15,19,20,17,26,25)$ |
| 6A-1 | $6^{4},1^{2}$ | $169$ | $6$ | $20$ | $( 1, 6,13, 2,10, 3)( 4, 5, 9,12,11, 7)(14,21,23,18,24,22)(15,25,26,17,20,19)$ |
| 8A1 | $8^{3},2$ | $169$ | $8$ | $22$ | $( 1,15, 4,21, 2,17,12,24)( 3,19, 7,14,13,26, 9,18)( 5,23,10,20,11,22, 6,25)( 8,16)$ |
| 8A-1 | $8^{3},2$ | $169$ | $8$ | $22$ | $( 1,24,12,17, 2,21, 4,15)( 3,18, 9,26,13,14, 7,19)( 5,25, 6,22,11,20,10,23)( 8,16)$ |
| 8A3 | $8^{3},2$ | $169$ | $8$ | $22$ | $( 1,21,12,15, 2,24, 4,17)( 3,14, 9,19,13,18, 7,26)( 5,20, 6,23,11,25,10,22)( 8,16)$ |
| 8A-3 | $8^{3},2$ | $169$ | $8$ | $22$ | $( 1,17, 4,24, 2,15,12,21)( 3,26, 7,18,13,19, 9,14)( 5,22,10,25,11,23, 6,20)( 8,16)$ |
| 12A1 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1, 9, 6,12,13,11, 2, 7,10, 4, 3, 5)(14,20,21,19,23,15,18,25,24,26,22,17)$ |
| 12A-1 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1, 5, 3, 4,10, 7, 2,11,13,12, 6, 9)(14,17,22,26,24,25,18,15,23,19,21,20)$ |
| 12A5 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1,11, 3,12,10, 9, 2, 5,13, 4, 6, 7)(14,15,22,19,24,20,18,17,23,26,21,25)$ |
| 12A-5 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1, 7, 6, 4,13, 5, 2, 9,10,12, 3,11)(14,25,21,26,23,17,18,20,24,19,22,15)$ |
| 13A | $13^{2}$ | $24$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13B | $13^{2}$ | $24$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13C | $13^{2}$ | $24$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13D | $13^{2}$ | $24$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13E | $13,1^{13}$ | $24$ | $13$ | $12$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$ |
| 13F | $13^{2}$ | $24$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13G | $13^{2}$ | $24$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 24A1 | $24,2$ | $169$ | $24$ | $24$ | $( 1,18, 5,15, 3,23, 4,19,10,21, 7,20, 2,14,11,17,13,22,12,26, 6,24, 9,25)( 8,16)$ |
| 24A-1 | $24,2$ | $169$ | $24$ | $24$ | $( 1,22, 7,15, 6,14, 4,25,13,21, 5,26, 2,23, 9,17,10,18,12,20, 3,24,11,19)( 8,16)$ |
| 24A5 | $24,2$ | $169$ | $24$ | $24$ | $( 1,25, 9,24, 6,26,12,22,13,17,11,14, 2,20, 7,21,10,19, 4,23, 3,15, 5,18)( 8,16)$ |
| 24A-5 | $24,2$ | $169$ | $24$ | $24$ | $( 1,14, 5,17, 3,22, 4,26,10,24, 7,25, 2,18,11,15,13,23,12,19, 6,21, 9,20)( 8,16)$ |
| 24A7 | $24,2$ | $169$ | $24$ | $24$ | $( 1,19,11,24, 3,20,12,18,10,17, 9,23, 2,26, 5,21,13,25, 4,14, 6,15, 7,22)( 8,16)$ |
| 24A-7 | $24,2$ | $169$ | $24$ | $24$ | $( 1,23, 7,17, 6,18, 4,20,13,24, 5,19, 2,22, 9,15,10,14,12,25, 3,21,11,26)( 8,16)$ |
| 24A11 | $24,2$ | $169$ | $24$ | $24$ | $( 1,26,11,21, 3,25,12,14,10,15, 9,22, 2,19, 5,24,13,20, 4,18, 6,17, 7,23)( 8,16)$ |
| 24A-11 | $24,2$ | $169$ | $24$ | $24$ | $( 1,20, 9,21, 6,19,12,23,13,15,11,18, 2,25, 7,24,10,26, 4,22, 3,17, 5,14)( 8,16)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
31 x 31 character tablemagma: CharacterTable(G);