Properties

Label 25T10
Order \(100\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:F_5$

Learn more about

Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $10$
Group :  $C_5:F_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,14,21)(2,20,13,24)(3,17,12,22)(4,19,11,25)(5,16,15,23)(6,9,10,7), (1,13,23,10,20)(2,14,24,6,16)(3,15,25,7,17)(4,11,21,8,18)(5,12,22,9,19)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$ x 2

Low degree siblings

10T10 x 2, 20T27 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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,16,22,12)( 7,18,21,15)( 8,20,25,13)( 9,17,24,11)(10,19,23,14)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,12,22,16)( 7,15,21,18)( 8,13,25,20)( 9,11,24,17)(10,14,23,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)(14,19) (15,18)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,15,18,22)( 2, 7,11,19,23)( 3, 8,12,20,24)( 4, 9,13,16,25) ( 5,10,14,17,21)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 7,12,16,21)( 2, 8,13,17,22)( 3, 9,14,18,23)( 4,10,15,19,24) ( 5, 6,11,20,25)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 8,14,19,25)( 2, 9,15,20,21)( 3,10,11,16,22)( 4, 6,12,17,23) ( 5, 7,13,18,24)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 9,11,17,24)( 2,10,12,18,25)( 3, 6,13,19,21)( 4, 7,14,20,22) ( 5, 8,15,16,23)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,10,13,20,23)( 2, 6,14,16,24)( 3, 7,15,17,25)( 4, 8,11,18,21) ( 5, 9,12,19,22)$

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [100, 12]
Character table:   
      2  2  2  2  2  .  .  .  .  .  .
      5  2  .  .  .  2  2  2  2  2  2

        1a 4a 4b 2a 5a 5b 5c 5d 5e 5f
     2P 1a 2a 2a 1a 5a 5c 5b 5f 5e 5d
     3P 1a 4b 4a 2a 5a 5c 5b 5f 5e 5d
     5P 1a 4a 4b 2a 1a 1a 1a 1a 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1  1  1  1  1  1  1
X.3      1  A -A -1  1  1  1  1  1  1
X.4      1 -A  A -1  1  1  1  1  1  1
X.5      4  .  .  .  4 -1 -1 -1 -1 -1
X.6      4  .  .  . -1 -1 -1 -1  4 -1
X.7      4  .  .  . -1  B *B  C -1 *C
X.8      4  .  .  . -1 *B  B *C -1  C
X.9      4  .  .  . -1  C *C *B -1  B
X.10     4  .  .  . -1 *C  C  B -1 *B

A = -E(4)
  = -Sqrt(-1) = -i
B = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
C = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4
  = (3+Sqrt(5))/2 = 2+b5