# Properties

 Label 25T10 Order $$100$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:F_5$

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$ x 2

## Low degree siblings

10T10 x 2, 20T27 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 1$ $25$ $4$ $( 2, 3, 5, 4)( 6,16,22,12)( 7,18,21,15)( 8,20,25,13)( 9,17,24,11)(10,19,23,14)$ $4, 4, 4, 4, 4, 4, 1$ $25$ $4$ $( 2, 4, 5, 3)( 6,12,22,16)( 7,15,21,18)( 8,13,25,20)( 9,11,24,17)(10,14,23,19)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)(14,19) (15,18)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 6,15,18,22)( 2, 7,11,19,23)( 3, 8,12,20,24)( 4, 9,13,16,25) ( 5,10,14,17,21)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 7,12,16,21)( 2, 8,13,17,22)( 3, 9,14,18,23)( 4,10,15,19,24) ( 5, 6,11,20,25)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 8,14,19,25)( 2, 9,15,20,21)( 3,10,11,16,22)( 4, 6,12,17,23) ( 5, 7,13,18,24)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1, 9,11,17,24)( 2,10,12,18,25)( 3, 6,13,19,21)( 4, 7,14,20,22) ( 5, 8,15,16,23)$ $5, 5, 5, 5, 5$ $4$ $5$ $( 1,10,13,20,23)( 2, 6,14,16,24)( 3, 7,15,17,25)( 4, 8,11,18,21) ( 5, 9,12,19,22)$

## Group invariants

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