Properties

Label 25T28
Order \(300\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:(C_3:C_4)$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $28$
Group :  $C_5^2:(C_3:C_4)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,13,9)(2,17,12,22)(3,4,11,15)(5,23,14,16)(6,25,8,19)(18,24,21,20), (1,7,25,19)(2,22,24,4)(3,12,23,14)(5,17,21,9)(6,10,20,16)(8,15,18,11)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T17 x 2, 30T71 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 $ $75$ $4$ $( 2, 3, 5, 4)( 6,19,21,13)( 7,16,25,11)( 8,18,24,14)( 9,20,23,12)(10,17,22,15)$
$ 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, 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 $ $50$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

Group invariants

Order:  $300=2^{2} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [300, 23]
Character table:   
     2  2  2  2  2  1  1  .  .
     3  1  .  .  1  1  1  .  .
     5  2  .  .  .  .  .  2  2

       1a 4a 4b 2a 3a 6a 5a 5b
    2P 1a 2a 2a 1a 3a 3a 5a 5b
    3P 1a 4b 4a 2a 1a 2a 5a 5b
    5P 1a 4a 4b 2a 3a 6a 1a 1a

X.1     1  1  1  1  1  1  1  1
X.2     1 -1 -1  1  1  1  1  1
X.3     1  A -A -1  1 -1  1  1
X.4     1 -A  A -1  1 -1  1  1
X.5     2  .  . -2 -1  1  2  2
X.6     2  .  .  2 -1 -1  2  2
X.7    12  .  .  .  .  .  2 -3
X.8    12  .  .  .  .  . -3  2

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