Group action invariants
Degree $n$: | $25$ | |
Transitive number $t$: | $43$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,2,4,3)(6,20,24,15)(7,17,23,13)(8,19,22,11)(9,16,21,14)(10,18,25,12), (1,11)(2,17)(3,23)(5,10)(7,12)(8,18)(9,24)(14,19)(15,25)(16,21) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $24$: $S_3 \times C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Low degree siblings
15T27 x 2, 30T150 x 2, 30T153 x 2, 30T155 x 2Siblings are shown 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 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$ |
$ 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6) ( 5,23,16,14, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $50$ | $3$ | $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$ |
$ 6, 6, 6, 6, 1 $ | $50$ | $6$ | $( 2,21,25, 5, 6, 7)( 3,16,19, 4,11,13)( 8,22,20,24,10,12)( 9,17,14,23,15,18)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $75$ | $4$ | $( 2, 3, 5, 4)( 6,19,21,13)( 7,16,25,11)( 8,18,24,14)( 9,20,23,12)(10,17,22,15)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $75$ | $4$ | $( 2, 4, 5, 3)( 6,13,21,19)( 7,11,25,16)( 8,14,24,18)( 9,12,23,20)(10,15,22,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$ |
$ 10, 10, 5 $ | $60$ | $10$ | $( 1, 6, 5,10, 4, 9, 3, 8, 2, 7)(11,24,15,23,14,22,13,21,12,25)(16,18,20,17,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)$ |
$ 10, 10, 5 $ | $60$ | $10$ | $( 1, 6,12,17,23, 3, 9,14,20,25)( 2,10,13,16,24)( 4, 8,15,19,21, 5, 7,11,18,22)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$ |
$ 12, 12, 1 $ | $50$ | $12$ | $( 2,11, 7, 4, 6,19, 5,16,25, 3,21,13)( 8,14,12,17,10, 9,24,18,20,15,22,23)$ |
$ 12, 12, 1 $ | $50$ | $12$ | $( 2,16, 7, 3, 6,13, 5,11,25, 4,21,19)( 8,18,12,15,10,23,24,14,20,17,22, 9)$ |
Group invariants
Order: | $600=2^{3} \cdot 3 \cdot 5^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [600, 151] |
Character table: |
2 3 1 1 3 2 2 3 3 3 1 3 1 3 3 2 2 3 1 . . 1 1 1 . . . . . . 1 1 1 1 5 2 2 2 . . . . . 1 1 1 1 . . . . 1a 5a 5b 2a 3a 6a 4a 4b 2b 10a 2c 10b 4c 4d 12a 12b 2P 1a 5a 5b 1a 3a 3a 2a 2a 1a 5a 1a 5b 2a 2a 6a 6a 3P 1a 5a 5b 2a 1a 2a 4b 4a 2b 10a 2c 10b 4d 4c 4d 4c 5P 1a 1a 1a 2a 3a 6a 4a 4b 2b 2b 2c 2c 4c 4d 12a 12b 7P 1a 5a 5b 2a 3a 6a 4b 4a 2b 10a 2c 10b 4d 4c 12b 12a 11P 1a 5a 5b 2a 3a 6a 4b 4a 2b 10a 2c 10b 4d 4c 12b 12a 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 1 1 1 -1 1 -1 A -A -1 -1 1 1 -A A -A A X.6 1 1 1 -1 1 -1 -A A -1 -1 1 1 A -A A -A X.7 1 1 1 -1 1 -1 A -A 1 1 -1 -1 A -A A -A X.8 1 1 1 -1 1 -1 -A A 1 1 -1 -1 -A A -A A X.9 2 2 2 2 -1 -1 . . . . . . -2 -2 1 1 X.10 2 2 2 2 -1 -1 . . . . . . 2 2 -1 -1 X.11 2 2 2 -2 -1 1 . . . . . . B -B -A A X.12 2 2 2 -2 -1 1 . . . . . . -B B A -A X.13 12 -3 2 . . . . . -4 1 . . . . . . X.14 12 -3 2 . . . . . 4 -1 . . . . . . X.15 12 2 -3 . . . . . . . -4 1 . . . . X.16 12 2 -3 . . . . . . . 4 -1 . . . . A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i |