# Properties

 Label 25T38 Order $$500$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5.D_5^2$

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## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $38$ Group : $C_5.D_5^2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,4,2,5,3)(6,25,10,21,9,22,8,23,7,24)(11,18,14,20,12,17,15,19,13,16), (1,14,10,23,20,4,15,7,25,16)(2,11,9,22,17,3,13,8,21,19)(5,12,6,24,18) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$ x 2
20:  $D_{10}$ x 2
100:  $D_5^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $D_{5}$

## Low degree siblings

25T38

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

## Group invariants

 Order: $500=2^{2} \cdot 5^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [500, 27]
 Character table:  2 2 1 1 2 2 2 1 1 1 1 1 1 1 1 . . 1 1 . . 5 3 2 2 1 1 1 1 1 3 1 3 1 1 2 2 2 1 2 2 2 1a 5a 5b 2a 2b 2c 10a 10b 5c 10c 5d 10d 10e 5e 5f 5g 10f 5h 5i 5j 2P 1a 5b 5a 1a 1a 1a 5b 5a 5d 5d 5c 5c 5h 5h 5i 5j 5e 5e 5f 5g 3P 1a 5b 5a 2a 2b 2c 10b 10a 5d 10d 5c 10c 10f 5h 5i 5j 10e 5e 5f 5g 5P 1a 1a 1a 2a 2b 2c 2c 2c 1a 2a 1a 2a 2b 1a 1a 1a 2b 1a 1a 1a 7P 1a 5b 5a 2a 2b 2c 10b 10a 5d 10d 5c 10c 10f 5h 5i 5j 10e 5e 5f 5g X.1 1 1 1 1 1 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 1 1 -1 1 1 1 X.3 1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 1 1 1 1 1 1 X.4 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 1 1 1 X.5 2 2 2 . -2 . . . 2 . 2 . -*A *A *A *A -A A A A X.6 2 2 2 . -2 . . . 2 . 2 . -A A A A -*A *A *A *A X.7 2 2 2 . 2 . . . 2 . 2 . A A A A *A *A *A *A X.8 2 2 2 . 2 . . . 2 . 2 . *A *A *A *A A A A A X.9 2 A *A . . -2 -A -*A 2 . 2 . . 2 *A A . 2 A *A X.10 2 *A A . . -2 -*A -A 2 . 2 . . 2 A *A . 2 *A A X.11 2 A *A . . 2 A *A 2 . 2 . . 2 *A A . 2 A *A X.12 2 *A A . . 2 *A A 2 . 2 . . 2 A *A . 2 *A A X.13 4 B *B . . . . . 4 . 4 . . B -1 *D . *B -1 D X.14 4 *B B . . . . . 4 . 4 . . *B -1 D . B -1 *D X.15 4 B *B . . . . . 4 . 4 . . *B D -1 . B *D -1 X.16 4 *B B . . . . . 4 . 4 . . B *D -1 . *B D -1 X.17 10 . . -2 . . . . C -A *C -*A . . . . . . . . X.18 10 . . -2 . . . . *C -*A C -A . . . . . . . . X.19 10 . . 2 . . . . C A *C *A . . . . . . . . X.20 10 . . 2 . . . . *C *A C A . . . . . . . . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 C = 5*E(5)^2+5*E(5)^3 = (-5-5*Sqrt(5))/2 = -5-5b5 D = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4 = (3-Sqrt(5))/2 = 1-b5