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Properties

Label 25T8
Order \(100\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{25}.C_2$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $8$
Group :  $D_{25}.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,7,23)(2,14,6,21)(3,11,10,24)(4,13,9,22)(5,15,8,25)(16,20,18,19), (1,14)(2,13)(3,12)(4,11)(5,15)(6,8)(9,10)(16,23)(17,22)(18,21)(19,25)(20,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

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

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [100, 3]
Character table:    
      2  2  2  2  2  .   .   .   .   .   .
      5  2  .  .  .  2   2   2   2   2   2

        1a 4a 4b 2a 5a 25a 25b 25c 25d 25e
     2P 1a 2a 2a 1a 5a 25e 25a 25b 25c 25d
     3P 1a 4b 4a 2a 5a 25d 25e 25a 25b 25c
     5P 1a 4a 4b 2a 1a  5a  5a  5a  5a  5a
     7P 1a 4b 4a 2a 5a 25a 25b 25c 25d 25e
    11P 1a 4b 4a 2a 5a 25e 25a 25b 25c 25d
    13P 1a 4a 4b 2a 5a 25b 25c 25d 25e 25a
    17P 1a 4a 4b 2a 5a 25c 25d 25e 25a 25b
    19P 1a 4b 4a 2a 5a 25c 25d 25e 25a 25b
    23P 1a 4b 4a 2a 5a 25e 25a 25b 25c 25d

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   B   D   C   F   E
X.7      4  .  .  . -1   C   F   E   B   D
X.8      4  .  .  . -1   D   C   F   E   B
X.9      4  .  .  . -1   E   B   D   C   F
X.10     4  .  .  . -1   F   E   B   D   C

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(25)^4-E(25)^6+E(25)^7-E(25)^9-E(25)^11-E(25)^14-E(25)^16+E(25)^18-E(25)^19-E(25)^21
C = E(25)^6+E(25)^8+E(25)^17+E(25)^19
D = E(25)^9+E(25)^12+E(25)^13+E(25)^16
E = -E(25)^3-E(25)^7-E(25)^8+E(25)^11-E(25)^12-E(25)^13+E(25)^14-E(25)^17-E(25)^18-E(25)^22
F = E(25)^3+E(25)^4+E(25)^21+E(25)^22