Group action invariants
| Degree $n$: | $26$ | |
| Transitive number $t$: | $49$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,22,8,25,5,23)(2,26,13,21,4,16)(3,24,9,17,11,19)(6,18)(7,20,10,14,12,15), (1,24,12,23,3,17,10,25,13,21,5,18)(2,20)(4,14,8,16,11,26)(6,15,7,19)(9,22) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ 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
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $117$ | $2$ | $( 1, 5)( 2, 7)( 3,12)(10,13)(14,24)(15,19)(16,21)(17,26)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ | $702$ | $4$ | $( 1,13, 5,10)( 2, 3, 7,12)( 4, 9)( 8,11)(14,17,24,26)(15,21,19,16)(18,25) (22,23)$ |
| $ 8, 8, 4, 4, 2 $ | $1404$ | $8$ | $( 1,15,13,21, 5,19,10,16)( 2,24, 3,26, 7,14,12,17)( 4,18, 9,25)( 6,20) ( 8,22,11,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $234$ | $2$ | $( 1,18)( 2,19)( 3,17)( 4,21)( 5,16)( 6,24)( 7,23)( 8,15)( 9,26)(10,14)(11,25) (12,22)(13,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $624$ | $3$ | $( 1,11, 4)( 2,12, 9)( 5, 6,13)( 7, 8,10)(14,23,15)(16,24,20)(18,25,21) (19,22,26)$ |
| $ 6, 6, 6, 6, 2 $ | $1872$ | $6$ | $( 1,21,11,18, 4,25)( 2,26,12,19, 9,22)( 3,17)( 5,20, 6,16,13,24) ( 7,14, 8,23,10,15)$ |
| $ 13, 13 $ | $864$ | $13$ | $( 1, 5, 9, 8,12,11, 2, 7, 3, 6,10, 4,13)(14,22,17,23,25,16,21,15,18,19,20,26, 24)$ |
| $ 13, 13 $ | $864$ | $13$ | $( 1, 3, 8, 4, 2, 5, 6,12,13, 7, 9,10,11)(14,18,23,26,21,22,19,25,24,15,17,20, 16)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $104$ | $3$ | $( 1, 2,10)( 3, 6,12)( 5,13, 7)(14,25,24)(15,22,19)(17,18,26)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2 $ | $234$ | $4$ | $( 1,15)( 2,22)( 3,18,13,24)( 4,20)( 5,25,12,17)( 6,26, 7,14)( 8,21,11,23) ( 9,16)(10,19)$ |
| $ 6, 6, 3, 3, 2, 2, 1, 1, 1, 1 $ | $936$ | $6$ | $( 1,10, 2)( 3, 5, 6,13,12, 7)( 8,11)(14,18,25,26,24,17)(15,19,22)(21,23)$ |
| $ 12, 6, 4, 2, 2 $ | $936$ | $12$ | $( 1,22,10,15, 2,19)( 3,14, 5,18, 6,25,13,26,12,24, 7,17)( 4,20)( 8,23,11,21) ( 9,16)$ |
| $ 12, 6, 4, 2, 2 $ | $936$ | $12$ | $( 1,22,10,15, 2,19)( 3,26, 5,24, 6,17,13,14,12,18, 7,25)( 4,20)( 8,21,11,23) ( 9,16)$ |
| $ 8, 8, 4, 4, 1, 1 $ | $1404$ | $8$ | $( 2,12, 6,13, 4,11, 5, 9)( 3,10, 7, 8)(14,19,22,26,23,25,21,20)(15,16,24,18)$ |
Group invariants
| Order: | $11232=2^{5} \cdot 3^{3} \cdot 13$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | not available |
| Character table: |
2 5 4 1 1 2 5 4 3 4 2 2 2 . . 3
3 3 1 2 1 3 1 . . 1 1 1 1 . . .
13 1 . . . . . . . . . . . 1 1 .
1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12a 12b 13a 13b 8b
2P 1a 1a 3a 3a 3b 1a 2b 4a 2b 3b 6b 6b 13b 13a 4a
3P 1a 2a 1a 2a 1a 2b 4a 8a 4b 2b 4b 4b 13a 13b 8b
5P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12b 12a 13b 13a 8b
7P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12b 12a 13b 13a 8b
11P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12a 12b 13b 13a 8b
13P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12a 12b 1a 1a 8b
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1
X.3 12 . . . 3 4 . . . 1 A -A -1 -1 .
X.4 12 . . . 3 4 . . . 1 -A A -1 -1 .
X.5 13 -1 1 -1 4 -3 1 1 3 . . . . . -1
X.6 13 1 1 1 4 -3 1 -1 -3 . . . . . -1
X.7 26 -2 -1 1 -1 2 2 . 2 -1 -1 -1 . . .
X.8 26 2 -1 -1 -1 2 2 . -2 -1 1 1 . . .
X.9 27 -3 . . . 3 -1 1 -3 . . . 1 1 -1
X.10 27 3 . . . 3 -1 -1 3 . . . 1 1 -1
X.11 32 . 2 . -4 . . . . . . . B *B .
X.12 32 . 2 . -4 . . . . . . . *B B .
X.13 39 -3 . . 3 -1 -1 -1 1 -1 1 1 . . 1
X.14 39 3 . . 3 -1 -1 1 -1 -1 -1 -1 . . 1
X.15 52 . -2 . -2 -4 . . . 2 . . . . .
A = -E(12)^7+E(12)^11
= Sqrt(3) = r3
B = E(13)^2+E(13)^5+E(13)^6+E(13)^7+E(13)^8+E(13)^11
= (-1-Sqrt(13))/2 = -1-b13
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