Properties

Label 25T9
Order \(100\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_4$

Learn more about

Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $9$
Group :  $C_5^2:C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,24,6,17)(2,15,25,7,18)(3,11,21,8,19)(4,12,22,9,20)(5,13,23,10,16), (1,25,19,12,10)(2,21,20,13,6)(3,22,16,14,7)(4,23,17,15,8)(5,24,18,11,9), (1,19,25,10)(2,16,24,8)(3,18,23,6)(4,20,22,9)(5,17,21,7)(11,15,13,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$ x 6

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

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [100, 11]
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 5b 5c 5d 5e 5f
     3P 1a 4b 4a 2a 5a 5b 5c 5d 5e 5f
     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  4 -1 -1 -1 -1
X.7      4  .  .  . -1 -1  4 -1 -1 -1
X.8      4  .  .  . -1 -1 -1  4 -1 -1
X.9      4  .  .  . -1 -1 -1 -1 -1  4
X.10     4  .  .  . -1 -1 -1 -1  4 -1

A = -E(4)
  = -Sqrt(-1) = -i