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Properties

Label 25T45
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 $t$ :  $45$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,21,12,6,4,11,25)(2,19,24,20,10,18,13,17)(3,5,22,23,9,7,15,14), (1,14,20,24,12,4,23,19)(2,7,18,8,11,6,25,10)(3,5,16,17,15,13,22,21)
$|\Aut(F/K)|$:  $1$

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 5: None

Low degree siblings

30T143

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 $ $24$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $50$ $3$ $( 2,25, 6)( 3,19,11)( 4,13,16)( 5, 7,21)( 8,20,10)( 9,14,15)(12,22,24) (17,23,18)$
$ 6, 6, 6, 6, 1 $ $50$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$
$ 12, 12, 1 $ $50$ $12$ $( 2,13,21, 3,25,16, 5,19, 6, 4, 7,11)( 8,23,22,15,20,18,24, 9,10,17,12,14)$
$ 12, 12, 1 $ $50$ $12$ $( 2,19,21, 4,25,11, 5,13, 6, 3, 7,16)( 8, 9,22,17,20,14,24,23,10,15,12,18)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2,12, 4, 9, 5,20, 3,23)( 6,22,16,14,21,10,11,18)( 7, 8,19,17,25,24,13,15)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2,20, 4,23, 5,12, 3, 9)( 6,10,16,18,21,22,11,14)( 7,24,19,15,25, 8,13,17)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2, 9, 3,12, 5,23, 4,20)( 6,14,11,22,21,18,16,10)( 7,17,13, 8,25,15,19,24)$
$ 8, 8, 8, 1 $ $75$ $8$ $( 2,23, 3,20, 5, 9, 4,12)( 6,18,11,10,21,14,16,22)( 7,15,13,24,25,17,19, 8)$

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [600, 148]
Character table:    
      2  3  .  3  3  3  2  2   2   2   3   3   3   3
      3  1  .  1  1  1  1  1   1   1   .   .   .   .
      5  2  2  .  .  .  .  .   .   .   .   .   .   .

        1a 5a 2a 4a 4b 3a 6a 12a 12b  8a  8b  8c  8d
     2P 1a 5a 1a 2a 2a 3a 3a  6a  6a  4a  4a  4b  4b
     3P 1a 5a 2a 4b 4a 1a 2a  4b  4a  8c  8d  8a  8b
     5P 1a 1a 2a 4a 4b 3a 6a 12a 12b  8b  8a  8d  8c
     7P 1a 5a 2a 4b 4a 3a 6a 12b 12a  8d  8c  8b  8a
    11P 1a 5a 2a 4b 4a 3a 6a 12b 12a  8c  8d  8a  8b

X.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
X.3      1  1 -1  A -A  1 -1   A  -A   C  -C -/C  /C
X.4      1  1 -1  A -A  1 -1   A  -A  -C   C  /C -/C
X.5      1  1 -1 -A  A  1 -1  -A   A -/C  /C   C  -C
X.6      1  1 -1 -A  A  1 -1  -A   A  /C -/C  -C   C
X.7      1  1  1 -1 -1  1  1  -1  -1   A   A  -A  -A
X.8      1  1  1 -1 -1  1  1  -1  -1  -A  -A   A   A
X.9      2  2  2 -2 -2 -1 -1   1   1   .   .   .   .
X.10     2  2  2  2  2 -1 -1  -1  -1   .   .   .   .
X.11     2  2 -2  B -B -1  1  -A   A   .   .   .   .
X.12     2  2 -2 -B  B -1  1   A  -A   .   .   .   .
X.13    24 -1  .  .  .  .  .   .   .   .   .   .   .

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = -E(8)^3