# Properties

 Label 25T12 Order $$100$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_5^2$

## Group action invariants

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

## 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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $D_{5}$ x 2

## Low degree siblings

10T9 x 2, 20T28 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $5$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $5, 5, 5, 5, 5$ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $10$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$ $5, 5, 5, 5, 5$ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $10, 10, 5$ $10$ $10$ $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$ $5, 5, 5, 5, 5$ $2$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$ $10, 10, 5$ $10$ $10$ $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ $5, 5, 5, 5, 5$ $2$ $5$ $( 1,11,21, 6,16)( 2,12,22, 7,17)( 3,13,23, 8,18)( 4,14,24, 9,19) ( 5,15,25,10,20)$ $10, 10, 5$ $10$ $10$ $( 1,11,21, 6,16)( 2,15,22,10,17, 5,12,25, 7,20)( 3,14,23, 9,18, 4,13,24, 8,19)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1,13,25, 7,19)( 2,14,21, 8,20)( 3,15,22, 9,16)( 4,11,23,10,17) ( 5,12,24, 6,18)$

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [100, 13]
 Character table:  2 2 2 2 2 1 1 1 1 1 1 . . 1 1 . . 5 2 1 1 . 2 1 2 1 2 1 2 2 2 1 2 2 1a 2a 2b 2c 5a 10a 5b 10b 5c 10c 5d 5e 5f 10d 5g 5h 2P 1a 1a 1a 1a 5b 5b 5a 5a 5f 5f 5h 5g 5c 5c 5e 5d 3P 1a 2a 2b 2c 5b 10b 5a 10a 5f 10d 5h 5g 5c 10c 5e 5d 5P 1a 2a 2b 2c 1a 2a 1a 2a 1a 2b 1a 1a 1a 2b 1a 1a 7P 1a 2a 2b 2c 5b 10b 5a 10a 5f 10d 5h 5g 5c 10c 5e 5d X.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 X.3 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 X.5 2 -2 . . A -A *A -*A 2 . A *A 2 . A *A X.6 2 -2 . . *A -*A A -A 2 . *A A 2 . *A A X.7 2 . -2 . 2 . 2 . A -A A A *A -*A *A *A X.8 2 . -2 . 2 . 2 . *A -*A *A *A A -A A A X.9 2 . 2 . 2 . 2 . A A A A *A *A *A *A X.10 2 . 2 . 2 . 2 . *A *A *A *A A A A A X.11 2 2 . . A A *A *A 2 . A *A 2 . A *A X.12 2 2 . . *A *A A A 2 . *A A 2 . *A A X.13 4 . . . B . *B . B . C -1 *B . -1 *C X.14 4 . . . *B . B . *B . *C -1 B . -1 C X.15 4 . . . B . *B . *B . -1 *C B . C -1 X.16 4 . . . *B . B . B . -1 C *B . *C -1 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 = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4 = (3+Sqrt(5))/2 = 2+b5