# Properties

 Label 25T41 Order $$600$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No

## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $41$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,25,10,21,2,17)(3,14,22,24,13,5)(4,6,18,23,16,9)(7,15,19,20,12,8), (1,25,13,19)(2,11,12,3)(4,18,15,21)(5,9,14,10)(6,16,8,23)(17,24,22,20) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
12:  $A_4$
24:  $\SL(2,3)$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

30T126

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 5$ $24$ $5$ $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6) ( 5,23,16,14, 7)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $4, 4, 4, 4, 4, 4, 1$ $150$ $4$ $( 2, 4, 5, 3)( 6,14,21,18)( 7,12,25,20)( 8,15,24,17)( 9,13,23,19)(10,11,22,16)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $100$ $3$ $( 2,22, 9)( 3,18,12)( 4,14,20)( 5,10,23)( 6,19, 8)( 7,15,11)(13,24,21) (16,25,17)$ $6, 6, 6, 6, 1$ $100$ $6$ $( 2,10, 9, 5,22,23)( 3,14,12, 4,18,20)( 6,13, 8,21,19,24)( 7,17,11,25,15,16)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $100$ $3$ $( 2, 9,22)( 3,12,18)( 4,20,14)( 5,23,10)( 6, 8,19)( 7,11,15)(13,21,24) (16,17,25)$ $6, 6, 6, 6, 1$ $100$ $6$ $( 2,23,22, 5, 9,10)( 3,20,18, 4,12,14)( 6,24,19,21, 8,13)( 7,16,15,25,11,17)$

## Group invariants

 Order: $600=2^{3} \cdot 3 \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [600, 150]
 Character table:  2 3 . 3 2 1 1 1 1 3 1 . 1 . 1 1 1 1 5 2 2 . . . . . . 1a 5a 2a 4a 3a 6a 3b 6b 2P 1a 5a 1a 2a 3b 3b 3a 3a 3P 1a 5a 2a 4a 1a 2a 1a 2a 5P 1a 1a 2a 4a 3b 6b 3a 6a X.1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 A A /A /A X.3 1 1 1 1 /A /A A A X.4 2 2 -2 . -1 1 -1 1 X.5 2 2 -2 . -A A -/A /A X.6 2 2 -2 . -/A /A -A A X.7 3 3 3 -1 . . . . X.8 24 -1 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3