# Properties

 Label 25T6 Order $$75$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_5^2:C_3$

## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $6$ Group : $C_5^2:C_3$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (2,6,25)(3,11,19)(4,16,13)(5,21,7)(8,10,20)(9,15,14)(12,24,22)(17,18,23), (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) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

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

## Group invariants

 Order: $75=3 \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [75, 2]
 Character table:  3 1 1 1 . . . . . . . . 5 2 . . 2 2 2 2 2 2 2 2 1a 3a 3b 5a 5b 5c 5d 5e 5f 5g 5h 2P 1a 3b 3a 5b 5d 5a 5c 5f 5g 5h 5e 3P 1a 1a 1a 5c 5a 5d 5b 5h 5e 5f 5g 5P 1a 3b 3a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 1 1 1 1 1 1 X.3 1 /A A 1 1 1 1 1 1 1 1 X.4 3 . . B /C C /B D *D D *D X.5 3 . . C B /B /C *D D *D D X.6 3 . . /C /B B C *D D *D D X.7 3 . . /B C /C B D *D D *D X.8 3 . . D *D *D D C B /C /B X.9 3 . . D *D *D D /C /B C B X.10 3 . . *D D D *D /B C B /C X.11 3 . . *D D D *D B /C /B C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(5)+E(5)^3 C = 2*E(5)^3+E(5)^4 D = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5