# Properties

 Label 25T27 Degree $25$ Order $300$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_5^2:D_6$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $27$ Group: $C_5^2:D_6$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,17,5,22,6,23)(2,12,24,21,11,4)(3,7,18,25,16,10)(8,13,19,20,15,9), (1,11)(2,17)(3,23)(5,10)(7,12)(8,18)(9,24)(14,19)(15,25)(16,21)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

15T18 x 2, 30T66 x 2, 30T72 x 2, 30T80 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $15$ $2$ $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $15$ $2$ $( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $50$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$ $6, 6, 6, 6, 1$ $50$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $30$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,20,13,17,15,19,12,16,14,18)$ $10, 10, 5$ $30$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$ $10, 10, 5$ $30$ $10$ $( 1, 2, 8, 9,15,11,17,18,24,25)( 3,14,10,16,12,23,19, 5,21, 7)( 4,20, 6,22,13)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $10, 10, 5$ $30$ $10$ $( 1, 3,15,12,24,21, 8,10,17,19)( 2, 9,11,18,25)( 4,16,13, 5,22,14, 6,23,20, 7)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$

## Group invariants

 Order: $300=2^{2} \cdot 3 \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [300, 25]
 Character table:  2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 3 1 . . 1 1 1 . . . . . . . . 5 2 1 1 . . . 2 1 1 1 2 1 2 2 1a 2a 2b 2c 3a 6a 5a 10a 10b 10c 5b 10d 5c 5d 2P 1a 1a 1a 1a 3a 3a 5b 5b 5a 5c 5a 5d 5d 5c 3P 1a 2a 2b 2c 1a 2c 5b 10b 10a 10d 5a 10c 5d 5c 5P 1a 2a 2b 2c 3a 6a 1a 2a 2a 2b 1a 2b 1a 1a 7P 1a 2a 2b 2c 3a 6a 5b 10b 10a 10d 5a 10c 5d 5c X.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 X.3 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 X.5 2 . . -2 -1 1 2 . . . 2 . 2 2 X.6 2 . . 2 -1 -1 2 . . . 2 . 2 2 X.7 6 -2 . . . . A C *C . *A . *B B X.8 6 -2 . . . . *A *C C . A . B *B X.9 6 . -2 . . . B . . *C *B C A *A X.10 6 . -2 . . . *B . . C B *C *A A X.11 6 . 2 . . . B . . -*C *B -C A *A X.12 6 . 2 . . . *B . . -C B -*C *A A X.13 6 2 . . . . A -C -*C . *A . *B B X.14 6 2 . . . . *A -*C -C . A . B *B A = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4 = (-3-Sqrt(5))/2 = -2-b5 B = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 C = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5