Group action invariants
| Degree $n$: | $26$ | |
| Transitive number $t$: | $35$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18), (1,16,2,15)(3,14,13,17)(4,26,12,18)(5,25,11,19)(6,24,10,20)(7,23,9,21)(8,22) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ 8: $Q_8$ 12: $D_{6}$ 24: 24T5 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, 13 $ | $24$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $13$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 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 $ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,15, 2,21)( 3,14,13,22)( 4,20,12,16)( 5,26,11,23)( 6,19,10,17)( 7,25, 9,24) ( 8,18)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,21,24,19,23,25,26,20,17,22,18,16)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $338$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,15,12,26)( 2,16,11,25)( 3,17,10,24)( 4,18, 9,23)( 5,19, 8,22)( 6,20, 7,21) (13,14)$ |
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 2 2 2 2 2 2 2
3 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 3a 6a 4a 12a 12b 4b 4c
2P 1a 13b 13c 13a 13d 13g 13e 13f 1a 3a 3a 2a 6a 6a 2a 2a
3P 1a 13b 13c 13a 13d 13g 13e 13f 2a 1a 2a 4a 4b 4b 4b 4c
5P 1a 13a 13b 13c 13d 13e 13f 13g 2a 3a 6a 4a 12b 12a 4b 4c
7P 1a 13c 13a 13b 13d 13f 13g 13e 2a 3a 6a 4a 12b 12a 4b 4c
11P 1a 13b 13c 13a 13d 13g 13e 13f 2a 3a 6a 4a 12a 12b 4b 4c
13P 1a 1a 1a 1a 1a 1a 1a 1a 2a 3a 6a 4a 12a 12b 4b 4c
X.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
X.3 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1
X.4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1
X.5 2 2 2 2 2 2 2 2 2 -1 -1 . -1 -1 2 .
X.6 2 2 2 2 2 2 2 2 2 -1 -1 . 1 1 -2 .
X.7 2 2 2 2 2 2 2 2 -2 2 -2 . . . . .
X.8 2 2 2 2 2 2 2 2 -2 -1 1 . G -G . .
X.9 2 2 2 2 2 2 2 2 -2 -1 1 . -G G . .
X.10 24 -2 -2 -2 11 -2 -2 -2 . . . . . . . .
X.11 24 A C B -2 D E F . . . . . . . .
X.12 24 B A C -2 E F D . . . . . . . .
X.13 24 C B A -2 F D E . . . . . . . .
X.14 24 D F E -2 B C A . . . . . . . .
X.15 24 E D F -2 C A B . . . . . . . .
X.16 24 F E D -2 A B C . . . . . . . .
A = 2*E(13)+4*E(13)^2+4*E(13)^3+2*E(13)^5+2*E(13)^8+4*E(13)^10+4*E(13)^11+2*E(13)^12
B = 4*E(13)+2*E(13)^4+4*E(13)^5+2*E(13)^6+2*E(13)^7+4*E(13)^8+2*E(13)^9+4*E(13)^12
C = 2*E(13)^2+2*E(13)^3+4*E(13)^4+4*E(13)^6+4*E(13)^7+4*E(13)^9+2*E(13)^10+2*E(13)^11
D = -2*E(13)-3*E(13)^2-3*E(13)^3-2*E(13)^4-2*E(13)^5-2*E(13)^6-2*E(13)^7-2*E(13)^8-2*E(13)^9-3*E(13)^10-3*E(13)^11-2*E(13)^12
E = -3*E(13)-2*E(13)^2-2*E(13)^3-2*E(13)^4-3*E(13)^5-2*E(13)^6-2*E(13)^7-3*E(13)^8-2*E(13)^9-2*E(13)^10-2*E(13)^11-3*E(13)^12
F = -2*E(13)-2*E(13)^2-2*E(13)^3-3*E(13)^4-2*E(13)^5-3*E(13)^6-3*E(13)^7-2*E(13)^8-3*E(13)^9-2*E(13)^10-2*E(13)^11-2*E(13)^12
G = -E(12)^7+E(12)^11
= Sqrt(3) = r3
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