Properties

Label 25T43
Order \(600\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $43$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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