Show commands:
Magma
magma: G := TransitiveGroup(26, 38);
Group invariants
| Abstract group: | $D_{13}^2.D_4$ | magma: IdentifyGroup(G);
| |
| Order: | $5408=2^{5} \cdot 13^{2}$ | 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 $t$: | $38$ | 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: | $(1,21,3,18,5,15,7,25,9,22,11,19,13,16,2,26,4,23,6,20,8,17,10,14,12,24)$, $(1,13,12,11,10,9,8,7,6,5,4,3,2)(14,24,26,16)(15,19,25,21)(17,22,23,18)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $32$: $C_4\wr C_2$ 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 | Index | Representative |
| 1A | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{6},1^{14}$ | $26$ | $2$ | $6$ | $(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| 2B | $2^{13}$ | $52$ | $2$ | $13$ | $( 1,23)( 2,15)( 3,20)( 4,25)( 5,17)( 6,22)( 7,14)( 8,19)( 9,24)(10,16)(11,21)(12,26)(13,18)$ |
| 2C | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)$ |
| 4A1 | $4^{3},1^{14}$ | $26$ | $4$ | $9$ | $(14,21,25,18)(15,16,24,23)(17,19,22,20)$ |
| 4A-1 | $4^{3},1^{14}$ | $26$ | $4$ | $9$ | $(14,18,25,21)(15,23,24,16)(17,20,22,19)$ |
| 4B1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1,10, 4, 8)( 2, 5, 3,13)( 6,11,12, 7)(14,25,22,24)(15,20,21,16)(17,23,19,26)$ |
| 4B-1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 8, 4,10)( 2,13, 3, 5)( 6, 7,12,11)(14,24,22,25)(15,16,21,20)(17,26,19,23)$ |
| 4C | $4^{6},1^{2}$ | $338$ | $4$ | $18$ | $( 1, 3,13,11)( 2, 8,12, 6)( 4, 5,10, 9)(14,24,26,16)(15,19,25,21)(17,22,23,18)$ |
| 4D1 | $4^{3},2^{6},1^{2}$ | $338$ | $4$ | $15$ | $( 1, 3,13,11)( 2, 8,12, 6)( 4, 5,10, 9)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| 4D-1 | $4^{3},2^{6},1^{2}$ | $338$ | $4$ | $15$ | $( 1,11,13, 3)( 2, 6,12, 8)( 4, 9,10, 5)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| 4E | $4^{6},2$ | $676$ | $4$ | $19$ | $( 1,26, 3,24)( 2,25)( 4,23,13,14)( 5,22,12,15)( 6,21,11,16)( 7,20,10,17)( 8,19, 9,18)$ |
| 8A1 | $8^{3},2$ | $676$ | $8$ | $22$ | $( 1,26,10,17, 4,23, 8,19)( 2,25, 5,22, 3,24,13,14)( 6,21,11,16,12,15, 7,20)( 9,18)$ |
| 8A-1 | $8^{3},2$ | $676$ | $8$ | $22$ | $( 1,19, 8,23, 4,17,10,26)( 2,14,13,24, 3,22, 5,25)( 6,20, 7,15,12,16,11,21)( 9,18)$ |
| 13A1 | $13,1^{13}$ | $8$ | $13$ | $12$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$ |
| 13A2 | $13,1^{13}$ | $8$ | $13$ | $12$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$ |
| 13A4 | $13,1^{13}$ | $8$ | $13$ | $12$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$ |
| 13B1 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13B2 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13B4 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13C1 | $13^{2}$ | $32$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13C2 | $13^{2}$ | $32$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13C4 | $13^{2}$ | $32$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 26A1 | $13,2^{6},1$ | $104$ | $26$ | $18$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| 26A3 | $13,2^{6},1$ | $104$ | $26$ | $18$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| 26A7 | $13,2^{6},1$ | $104$ | $26$ | $18$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| 26B1 | $26$ | $208$ | $26$ | $25$ | $( 1,14,13,22,12,17,11,25,10,20, 9,15, 8,23, 7,18, 6,26, 5,21, 4,16, 3,24, 2,19)$ |
| 26B3 | $26$ | $208$ | $26$ | $25$ | $( 1,22,11,20, 8,18, 5,16, 2,14,12,25, 9,23, 6,21, 3,19,13,17,10,15, 7,26, 4,24)$ |
| 26B7 | $26$ | $208$ | $26$ | $25$ | $( 1,25, 7,16,13,20, 6,24,12,15, 5,19,11,23, 4,14,10,18, 3,22, 9,26, 2,17, 8,21)$ |
| 52A1 | $13,4^{3},1$ | $104$ | $52$ | $21$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,21,25,18)(15,16,24,23)(17,19,22,20)$ |
| 52A-1 | $13,4^{3},1$ | $104$ | $52$ | $21$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,25,21)(15,23,24,16)(17,20,22,19)$ |
| 52A3 | $13,4^{3},1$ | $104$ | $52$ | $21$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,18,25,21)(15,23,24,16)(17,20,22,19)$ |
| 52A-3 | $13,4^{3},1$ | $104$ | $52$ | $21$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,21,25,18)(15,16,24,23)(17,19,22,20)$ |
| 52A7 | $13,4^{3},1$ | $104$ | $52$ | $21$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,25,21)(15,23,24,16)(17,20,22,19)$ |
| 52A-7 | $13,4^{3},1$ | $104$ | $52$ | $21$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,21,25,18)(15,16,24,23)(17,19,22,20)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
35 x 35 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed