Group action invariants
Degree $n$: | $26$ | |
Transitive number $t$: | $42$ | |
Group: | $\PSL(2,25)$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,19,26,4,17,2,6,22,18,5,21,12,16)(3,8,7,14,23,13,24,10,25,9,11,15,20), (1,21,5,9,4,2,10,16,12,18,14,25,3)(6,23,7,24,17,11,8,20,26,15,22,19,13) |
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
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, 2, 2, 2, 2, 1, 1 $ | $325$ | $2$ | $( 1, 5)( 2, 3)( 4,26)( 6, 8)( 7,21)( 9,17)(10,19)(11,24)(12,20)(13,16)(14,23) (22,25)$ |
$ 4, 4, 4, 4, 4, 4, 1, 1 $ | $650$ | $4$ | $( 1,16, 5,13)( 2,20, 3,12)( 4, 7,26,21)( 6,22, 8,25)( 9,23,17,14)(10,24,19,11)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $650$ | $3$ | $( 1, 4, 2)( 3, 5,26)( 6,23,11)( 7,20,16)( 8,14,24)( 9,19,25)(10,22,17) (12,13,21)$ |
$ 6, 6, 6, 6, 1, 1 $ | $650$ | $6$ | $( 1, 3, 4, 5, 2,26)( 6,24,23, 8,11,14)( 7,13,20,21,16,12)( 9,22,19,17,25,10)$ |
$ 12, 12, 1, 1 $ | $650$ | $12$ | $( 1, 7, 3,13, 4,20, 5,21, 2,16,26,12)( 6,17,24,25,23,10, 8, 9,11,22,14,19)$ |
$ 12, 12, 1, 1 $ | $650$ | $12$ | $( 1,21, 3,16, 4,12, 5, 7, 2,13,26,20)( 6, 9,24,22,23,19, 8,17,11,25,14,10)$ |
$ 5, 5, 5, 5, 5, 1 $ | $312$ | $5$ | $( 1,24,22, 9,11)( 2,25,15,17, 4)( 3, 8,18,23,26)( 5,19, 6,14,10) ( 7,21,16,12,20)$ |
$ 5, 5, 5, 5, 5, 1 $ | $312$ | $5$ | $( 1, 5,18,16,15)( 2, 9,14, 3, 7)( 4,22, 6,26,20)( 8,21,25,11,10) (12,17,24,19,23)$ |
$ 13, 13 $ | $600$ | $13$ | $( 1,20,16, 9, 8, 4,15,26, 6,17,13,12, 5)( 2,14,18,23, 3, 7,19,25,11,24,22,10, 21)$ |
$ 13, 13 $ | $600$ | $13$ | $( 1, 6, 9,12,15,20,17, 8, 5,26,16,13, 4)( 2,11,23,10,19,14,24, 3,21,25,18,22, 7)$ |
$ 13, 13 $ | $600$ | $13$ | $( 1,17, 4,20,13,15,16,12,26, 9, 5, 6, 8)( 2,24, 7,14,22,19,18,10,25,23,21,11, 3)$ |
$ 13, 13 $ | $600$ | $13$ | $( 1,26,20, 6,16,17, 9,13, 8,12, 4, 5,15)( 2,25,14,11,18,24,23,22, 3,10, 7,21, 19)$ |
$ 13, 13 $ | $600$ | $13$ | $( 1, 9,15,17, 5,16, 4, 6,12,20, 8,26,13)( 2,23,19,24,21,18, 7,11,10,14, 3,25, 22)$ |
$ 13, 13 $ | $600$ | $13$ | $( 1,12,17,26, 4, 9,20, 5,13, 6,15, 8,16)( 2,10,24,25, 7,23,14,21,22,11,19, 3, 18)$ |
Group invariants
Order: | $7800=2^{3} \cdot 3 \cdot 5^{2} \cdot 13$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
GAP id: | not available |
Character table: |
2 3 . . . . . . 3 2 2 2 2 2 . . 3 1 . . . . . . 1 1 1 1 1 1 . . 5 2 . . . . . . . . . . . . 2 2 13 1 1 1 1 1 1 1 . . . . . . . . 1a 13a 13b 13c 13d 13e 13f 2a 3a 6a 4a 12a 12b 5a 5b 2P 1a 13f 13e 13b 13a 13d 13c 1a 3a 3a 2a 6a 6a 5a 5b 3P 1a 13e 13f 13a 13b 13c 13d 2a 1a 2a 4a 4a 4a 5a 5b 5P 1a 13b 13a 13d 13c 13f 13e 2a 3a 6a 4a 12b 12a 1a 1a 7P 1a 13d 13c 13f 13e 13b 13a 2a 3a 6a 4a 12b 12a 5a 5b 11P 1a 13f 13e 13b 13a 13d 13c 2a 3a 6a 4a 12a 12b 5a 5b 13P 1a 1a 1a 1a 1a 1a 1a 2a 3a 6a 4a 12a 12b 5a 5b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 13 . . . . . . 1 1 1 -1 -1 -1 3 -2 X.3 13 . . . . . . 1 1 1 -1 -1 -1 -2 3 X.4 24 A F E B C D . . . . . . -1 -1 X.5 24 B E D C F A . . . . . . -1 -1 X.6 24 C D A F E B . . . . . . -1 -1 X.7 24 D C F A B E . . . . . . -1 -1 X.8 24 E B C D A F . . . . . . -1 -1 X.9 24 F A B E D C . . . . . . -1 -1 X.10 25 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 . . X.11 26 . . . . . . 2 -1 -1 -2 1 1 1 1 X.12 26 . . . . . . 2 -1 -1 2 -1 -1 1 1 X.13 26 . . . . . . -2 2 -2 . . . 1 1 X.14 26 . . . . . . -2 -1 1 . G -G 1 1 X.15 26 . . . . . . -2 -1 1 . -G G 1 1 A = -E(13)-E(13)^12 B = -E(13)^6-E(13)^7 C = -E(13)^3-E(13)^10 D = -E(13)^2-E(13)^11 E = -E(13)^4-E(13)^9 F = -E(13)^5-E(13)^8 G = -E(12)^7+E(12)^11 = Sqrt(3) = r3 |