# Properties

 Label 25T28 Degree $25$ Order $300$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_5^2:(C_3:C_4)$

## 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) $|\Aut(F/K)|$: $1$ 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)

## 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 $\leq 47$

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

## 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