Show commands:
Magma
magma: G := TransitiveGroup(26, 36);
Group invariants
| Abstract group: | $C_{13}^2:D_{12}$ | 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 $t$: | $36$ | 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: | $(1,25,8,23,2,21,9,19,3,17,10,15,4,26,11,24,5,22,12,20,6,18,13,16,7,14)$, $(1,20,11,18,8,16,5,14,2,25,12,23,9,21,6,19,3,17,13,15,10,26,7,24,4,22)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $24$: $D_{12}$ 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 | $2^{13}$ | $78$ | $2$ | $13$ | $( 1,17)( 2,22)( 3,14)( 4,19)( 5,24)( 6,16)( 7,21)( 8,26)( 9,18)(10,23)(11,15)(12,20)(13,25)$ |
| 2B | $2^{13}$ | $78$ | $2$ | $13$ | $( 1,22)( 2,18)( 3,14)( 4,23)( 5,19)( 6,15)( 7,24)( 8,20)( 9,16)(10,25)(11,21)(12,17)(13,26)$ |
| 2C | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 6)( 2, 5)( 3, 4)( 7,13)( 8,12)( 9,11)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| 3A | $3^{8},1^{2}$ | $338$ | $3$ | $16$ | $( 1, 9, 7)( 2,12, 3)( 4, 5, 8)( 6,11,13)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$ |
| 4A | $4^{6},1^{2}$ | $338$ | $4$ | $18$ | $( 1, 4, 6, 3)( 2, 9, 5,11)( 7, 8,13,12)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| 6A | $6^{4},1^{2}$ | $338$ | $6$ | $20$ | $( 1,13, 9, 6, 7,11)( 2, 4,12, 5, 3, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| 12A1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 5,13, 3, 9, 8, 6, 2, 7, 4,11,12)(15,21,24,19,23,25,26,20,17,22,18,16)$ |
| 12A5 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 8,11, 3, 7, 5, 6,12, 9, 4,13, 2)(15,25,18,19,17,21,26,16,23,22,24,20)$ |
| 13A1 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13A2 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13A3 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13A4 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13A5 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13A6 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13B1 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13B2 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13B3 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13B4 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13B5 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13B6 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13C | $13,1^{13}$ | $24$ | $13$ | $12$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$ |
| 26A1 | $26$ | $156$ | $26$ | $25$ | $( 1,20,12,23,10,26, 8,16, 6,19, 4,22, 2,25,13,15,11,18, 9,21, 7,24, 5,14, 3,17)$ |
| 26A3 | $26$ | $156$ | $26$ | $25$ | $( 1,18, 9,19, 4,20,12,21, 7,22, 2,23,10,24, 5,25,13,26, 8,14, 3,15,11,16, 6,17)$ |
| 26A5 | $26$ | $156$ | $26$ | $25$ | $( 1,19, 4,21, 7,23,10,25,13,14, 3,16, 6,18, 9,20,12,22, 2,24, 5,26, 8,15,11,17)$ |
| 26A7 | $26$ | $156$ | $26$ | $25$ | $( 1,22, 2,14, 3,19, 4,24, 5,16, 6,21, 7,26, 8,18, 9,23,10,15,11,20,12,25,13,17)$ |
| 26A9 | $26$ | $156$ | $26$ | $25$ | $( 1,14, 3,19, 5,24, 7,16, 9,21,11,26,13,18, 2,23, 4,15, 6,20, 8,25,10,17,12,22)$ |
| 26A11 | $26$ | $156$ | $26$ | $25$ | $( 1,26,13,17,12,21,11,25,10,16, 9,20, 8,24, 7,15, 6,19, 5,23, 4,14, 3,18, 2,22)$ |
| 26B1 | $26$ | $156$ | $26$ | $25$ | $( 1,24, 7,26,13,15, 6,17,12,19, 5,21,11,23, 4,25,10,14, 3,16, 9,18, 2,20, 8,22)$ |
| 26B3 | $26$ | $156$ | $26$ | $25$ | $( 1,25,10,15, 6,18, 2,21,11,24, 7,14, 3,17,12,20, 8,23, 4,26,13,16, 9,19, 5,22)$ |
| 26B5 | $26$ | $156$ | $26$ | $25$ | $( 1,24, 5,18, 9,25,13,19, 4,26, 8,20,12,14, 3,21, 7,15,11,22, 2,16, 6,23,10,17)$ |
| 26B7 | $26$ | $156$ | $26$ | $25$ | $( 1,23, 4,24, 7,25,10,26,13,14, 3,15, 6,16, 9,17,12,18, 2,19, 5,20, 8,21,11,22)$ |
| 26B9 | $26$ | $156$ | $26$ | $25$ | $( 1,16, 9,23, 4,17,12,24, 7,18, 2,25,10,19, 5,26,13,20, 8,14, 3,21,11,15, 6,22)$ |
| 26B11 | $26$ | $156$ | $26$ | $25$ | $( 1,21, 7,25,13,16, 6,20,12,24, 5,15,11,19, 4,23,10,14, 3,18, 9,22, 2,26, 8,17)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
34 x 34 character tablemagma: CharacterTable(G);