# Properties

 Label 25T33 Degree $25$ Order $500$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^2:F_5$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $33$ Group: $C_5^2:F_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,8,12,17,25)(2,9,13,18,21)(3,10,14,19,22)(4,6,15,20,23)(5,7,11,16,24), (1,18,21,6)(2,17,22,10)(3,16,23,9)(4,20,24,8)(5,19,25,7)(11,13)(14,15), (1,8,5,7,4,6,3,10,2,9)(11,22,15,21,14,25,13,24,12,23)(16,18,20,17,19)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$ x 6
$100$:  25T9

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$

## Low degree siblings

25T33 x 5

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 1, 1, 1, 1, 1$ $20$ $5$ $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $25$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,17)(12,18)(13,19)(14,20)(15,16)$ $4, 4, 4, 4, 4, 2, 2, 1$ $125$ $4$ $( 2, 5)( 3, 4)( 6,11,25,20)( 7,15,21,19)( 8,14,22,18)( 9,13,23,17) (10,12,24,16)$ $4, 4, 4, 4, 4, 2, 2, 1$ $125$ $4$ $( 2, 5)( 3, 4)( 6,16,22,13)( 7,20,23,12)( 8,19,24,11)( 9,18,25,15) (10,17,21,14)$ $5, 5, 5, 5, 5$ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $50$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,16,13,18,15,20,12,17,14,19)$ $5, 5, 5, 5, 5$ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $10, 10, 5$ $50$ $10$ $( 1, 3, 5, 2, 4)( 6,21,10,25, 9,24, 8,23, 7,22)(11,20,15,19,14,18,13,17,12,16)$ $5, 5, 5, 5, 5$ $20$ $5$ $( 1, 6,11,20,25)( 2, 7,12,16,21)( 3, 8,13,17,22)( 4, 9,14,18,23) ( 5,10,15,19,24)$ $5, 5, 5, 5, 5$ $20$ $5$ $( 1, 6,12,18,21)( 2, 7,13,19,22)( 3, 8,14,20,23)( 4, 9,15,16,24) ( 5,10,11,17,25)$ $5, 5, 5, 5, 5$ $20$ $5$ $( 1, 6,13,16,22)( 2, 7,14,17,23)( 3, 8,15,18,24)( 4, 9,11,19,25) ( 5,10,12,20,21)$ $5, 5, 5, 5, 5$ $20$ $5$ $( 1, 6,14,19,23)( 2, 7,15,20,24)( 3, 8,11,16,25)( 4, 9,12,17,21) ( 5,10,13,18,22)$ $5, 5, 5, 5, 5$ $20$ $5$ $( 1, 6,15,17,24)( 2, 7,11,18,25)( 3, 8,12,19,21)( 4, 9,13,20,22) ( 5,10,14,16,23)$

## Group invariants

 Order: $500=2^{2} \cdot 5^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [500, 23]
 Character table:  2 2 . 2 2 2 1 1 1 1 . . . . . 5 3 2 1 . . 3 1 3 1 2 2 2 2 2 1a 5a 2a 4a 4b 5b 10a 5c 10b 5d 5e 5f 5g 5h 2P 1a 5a 1a 2a 2a 5c 5c 5b 5b 5d 5e 5f 5g 5h 3P 1a 5a 2a 4b 4a 5c 10b 5b 10a 5d 5e 5f 5g 5h 5P 1a 1a 2a 4a 4b 1a 2a 1a 2a 1a 1a 1a 1a 1a 7P 1a 5a 2a 4b 4a 5c 10b 5b 10a 5d 5e 5f 5g 5h X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 X.3 1 1 -1 A -A 1 -1 1 -1 1 1 1 1 1 X.4 1 1 -1 -A A 1 -1 1 -1 1 1 1 1 1 X.5 4 4 . . . 4 . 4 . -1 -1 -1 -1 -1 X.6 4 -1 . . . 4 . 4 . 4 -1 -1 -1 -1 X.7 4 -1 . . . 4 . 4 . -1 4 -1 -1 -1 X.8 4 -1 . . . 4 . 4 . -1 -1 4 -1 -1 X.9 4 -1 . . . 4 . 4 . -1 -1 -1 -1 4 X.10 4 -1 . . . 4 . 4 . -1 -1 -1 4 -1 X.11 10 . -2 . . B C *B *C . . . . . X.12 10 . -2 . . *B *C B C . . . . . X.13 10 . 2 . . B -C *B -*C . . . . . X.14 10 . 2 . . *B -*C B -C . . . . . A = -E(4) = -Sqrt(-1) = -i B = 5*E(5)^2+5*E(5)^3 = (-5-5*Sqrt(5))/2 = -5-5b5 C = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5