Group action invariants
| Degree $n$ : | $26$ | |
| Transitive number $t$ : | $31$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,5,24,12,20,8,26)(2,22,10,23,11,15,3,14)(4,19,7,21,9,18,6,16)(13,25), (1,15,5,14,11,19,7,20)(2,18,13,25,10,16,12,22)(3,21,8,23,9,26,4,24)(6,17) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 6: $S_3$ 8: $C_8$ 12: $C_3 : C_4$ 24: 24T8 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 $ | $1$ | $1$ | $()$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,21,24,19,23,25,26,20,17,22,18,16)$ |
| $ 8, 8, 8, 2 $ | $507$ | $8$ | $( 1,17, 5,24,12,20, 8,26)( 2,22,10,23,11,15, 3,14)( 4,19, 7,21, 9,18, 6,16) (13,25)$ |
| $ 8, 8, 8, 2 $ | $507$ | $8$ | $( 1,24, 3,14,13,16,11,26)( 2,19, 8,15,12,21, 6,25)( 4,22, 5,17,10,18, 9,23) ( 7,20)$ |
| $ 8, 8, 8, 2 $ | $507$ | $8$ | $( 1,16, 3,14, 6,24, 4,26)( 2,15,11,19, 5,25, 9,21)( 7,23,12,18,13,17, 8,22) (10,20)$ |
| $ 8, 8, 8, 2 $ | $507$ | $8$ | $( 1,25, 6,17, 7,18, 2,26)( 3,14, 9,20, 5,16,12,23)( 4,15)( 8,19,10,21,13,24, 11,22)$ |
Group invariants
| Order: | $4056=2^{3} \cdot 3 \cdot 13^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: |
2 3 . . . . . . . 3 3 3 2 2 2 2 3 3 3 3
3 1 . . . . . . . 1 1 1 1 1 1 1 . . . .
13 2 2 2 2 2 2 2 2 . . . . . . . . . . .
1a 13a 13b 13c 13d 13e 13f 13g 2a 4a 4b 3a 6a 12a 12b 8a 8b 8c 8d
2P 1a 13a 13f 13b 13g 13d 13c 13e 1a 2a 2a 3a 3a 6a 6a 4a 4a 4b 4b
3P 1a 13a 13f 13b 13g 13d 13c 13e 2a 4b 4a 1a 2a 4b 4a 8c 8d 8a 8b
5P 1a 13a 13b 13c 13d 13e 13f 13g 2a 4a 4b 3a 6a 12a 12b 8b 8a 8d 8c
7P 1a 13a 13c 13f 13e 13g 13b 13d 2a 4b 4a 3a 6a 12b 12a 8d 8c 8b 8a
11P 1a 13a 13f 13b 13g 13d 13c 13e 2a 4b 4a 3a 6a 12b 12a 8c 8d 8a 8b
13P 1a 1a 1a 1a 1a 1a 1a 1a 2a 4a 4b 3a 6a 12a 12b 8b 8a 8d 8c
X.1 1 1 1 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 -1 -1 -1 -1
X.3 1 1 1 1 1 1 1 1 -1 D -D 1 -1 D -D F -F -/F /F
X.4 1 1 1 1 1 1 1 1 -1 D -D 1 -1 D -D -F F /F -/F
X.5 1 1 1 1 1 1 1 1 -1 -D D 1 -1 -D D -/F /F F -F
X.6 1 1 1 1 1 1 1 1 -1 -D D 1 -1 -D D /F -/F -F F
X.7 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 D D -D -D
X.8 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -D -D D D
X.9 2 2 2 2 2 2 2 2 2 -2 -2 -1 -1 1 1 . . . .
X.10 2 2 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 . . . .
X.11 2 2 2 2 2 2 2 2 -2 E -E -1 1 -D D . . . .
X.12 2 2 2 2 2 2 2 2 -2 -E E -1 1 D -D . . . .
X.13 24 11 -2 -2 -2 -2 -2 -2 . . . . . . . . . . .
X.14 24 -2 A B -2 -2 C -2 . . . . . . . . . . .
X.15 24 -2 B C -2 -2 A -2 . . . . . . . . . . .
X.16 24 -2 C A -2 -2 B -2 . . . . . . . . . . .
X.17 24 -2 -2 -2 A B -2 C . . . . . . . . . . .
X.18 24 -2 -2 -2 B C -2 A . . . . . . . . . . .
X.19 24 -2 -2 -2 C A -2 B . . . . . . . . . . .
A = -4*E(13)^2-4*E(13)^3-3*E(13)^4-3*E(13)^6-3*E(13)^7-3*E(13)^9-4*E(13)^10-4*E(13)^11
B = -4*E(13)-3*E(13)^2-3*E(13)^3-4*E(13)^5-4*E(13)^8-3*E(13)^10-3*E(13)^11-4*E(13)^12
C = -3*E(13)-4*E(13)^4-3*E(13)^5-4*E(13)^6-4*E(13)^7-3*E(13)^8-4*E(13)^9-3*E(13)^12
D = -E(4)
= -Sqrt(-1) = -i
E = -2*E(4)
= -2*Sqrt(-1) = -2i
F = -E(8)^3
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