# Properties

 Label 25T36 Order $$500$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5^2:C_{10}.C_2$

## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $36$ Group : $C_5^2:C_{10}.C_2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,10,4,6)(2,7,3,9)(5,8)(11,15,12,13)(16,25,19,22)(17,24,18,23)(20,21), (1,23,2,21)(3,24,5,25)(4,22)(6,13)(7,14,10,12)(8,15,9,11)(16,18,19,17) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  $F_5$, 20T2
100:  20T26

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $D_{5}$

## Low degree siblings

25T39

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

## Group invariants

 Order: $500=2^{2} \cdot 5^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [500, 21]
 Character table:  2 2 . 2 2 2 . 1 1 . . 1 1 . . 5 3 2 . . 1 3 1 2 2 2 1 2 2 2 1a 5a 4a 4b 2a 5b 10a 5c 5d 5e 10b 5f 5g 5h 2P 1a 5a 2a 2a 1a 5b 5f 5f 5h 5g 5c 5c 5d 5e 3P 1a 5a 4b 4a 2a 5b 10b 5f 5g 5h 10a 5c 5e 5d 5P 1a 1a 4a 4b 2a 1a 2a 1a 1a 1a 2a 1a 1a 1a 7P 1a 5a 4b 4a 2a 5b 10b 5f 5h 5g 10a 5c 5d 5e 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 A -A -1 1 -1 1 1 1 -1 1 1 1 X.4 1 1 -A A -1 1 -1 1 1 1 -1 1 1 1 X.5 2 2 . . -2 2 B -B -B -B *B -*B -*B -*B X.6 2 2 . . -2 2 *B -*B -*B -*B B -B -B -B X.7 2 2 . . 2 2 -*B -*B -*B -*B -B -B -B -B X.8 2 2 . . 2 2 -B -B -B -B -*B -*B -*B -*B X.9 4 -1 . . . 4 . 4 -1 -1 . 4 -1 -1 X.10 4 -1 . . . 4 . C D /D . *C /E E X.11 4 -1 . . . 4 . C /D D . *C E /E X.12 4 -1 . . . 4 . *C E /E . C D /D X.13 4 -1 . . . 4 . *C /E E . C /D D X.14 20 . . . . -5 . . . . . . . . A = -E(4) = -Sqrt(-1) = -i B = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 C = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 D = -E(5)-E(5)^3+E(5)^4 E = -E(5)-E(5)^2+E(5)^3