# Properties

 Label 25T15 Degree $25$ Order $150$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_5^2:C_6$

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## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $15$ Group: $C_5^2:C_6$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,19,16)(2,24,15)(3,4,9)(5,14,22)(6,13,17)(7,18,11)(8,23,10)(20,21,25), (1,11,14,2,17,19)(3,23,24,5,10,9)(6,15,8,22,18,25)(7,16,13,21,12,20)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

15T12 x 2, 30T35 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $25$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$ $6, 6, 6, 6, 1$ $25$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$ $6, 6, 6, 6, 1$ $25$ $6$ $( 2,21,25, 5, 6, 7)( 3,16,19, 4,11,13)( 8,22,20,24,10,12)( 9,17,14,23,15,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $25$ $3$ $( 2,25, 6)( 3,19,11)( 4,13,16)( 5, 7,21)( 8,20,10)( 9,14,15)(12,22,24) (17,23,18)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$

## Group invariants

 Order: $150=2 \cdot 3 \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [150, 6]
 Character table:  2 1 1 1 1 1 1 . . . . 3 1 1 1 1 1 1 . . . . 5 2 . . . . . 2 2 2 2 1a 2a 3a 6a 6b 3b 5a 5b 5c 5d 2P 1a 1a 3b 3a 3b 3a 5b 5a 5d 5c 3P 1a 2a 1a 2a 2a 1a 5b 5a 5d 5c 5P 1a 2a 3b 6b 6a 3a 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 -1 A -/A -A /A 1 1 1 1 X.4 1 -1 /A -A -/A A 1 1 1 1 X.5 1 1 A /A A /A 1 1 1 1 X.6 1 1 /A A /A A 1 1 1 1 X.7 6 . . . . . B *B C *C X.8 6 . . . . . *B B *C C X.9 6 . . . . . C *C *B B X.10 6 . . . . . *C C B *B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -2*E(5)-2*E(5)^4 = 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