# Properties

 Label 25T26 Degree $25$ Order $300$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_5^2:C_{12}$

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $26$ Group: $C_5^2:C_{12}$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,4,9,22,21,11,10,7,2,14,15,25)(3,24,16,23,6,17,8,12,20,13,5,19), (1,2)(3,5)(6,22)(7,21)(8,25)(9,24)(10,23)(11,17)(12,16)(13,20)(14,19)(15,18)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

15T19 x 2, 30T78 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 1$ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$ $4, 4, 4, 4, 4, 4, 1$ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $25$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$ $6, 6, 6, 6, 1$ $25$ $6$ $( 2, 7, 6, 5,25,21)( 3,13,11, 4,19,16)( 8,12,10,24,20,22)( 9,18,15,23,14,17)$ $12, 12, 1$ $25$ $12$ $( 2,11, 7, 4, 6,19, 5,16,25, 3,21,13)( 8,14,12,17,10, 9,24,18,20,15,22,23)$ $12, 12, 1$ $25$ $12$ $( 2,13,21, 3,25,16, 5,19, 6, 4, 7,11)( 8,23,22,15,20,18,24, 9,10,17,12,14)$ $12, 12, 1$ $25$ $12$ $( 2,16, 7, 3, 6,13, 5,11,25, 4,21,19)( 8,18,12,15,10,23,24,14,20,17,22, 9)$ $12, 12, 1$ $25$ $12$ $( 2,19,21, 4,25,11, 5,13, 6, 3, 7,16)( 8, 9,22,17,20,14,24,23,10,15,12,18)$ $6, 6, 6, 6, 1$ $25$ $6$ $( 2,21,25, 5, 6, 7)( 3,16,19, 4,11,13)( 8,22,20,24,10,12)( 9,17,14,23,15,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $25$ $3$ $( 2,25, 6)( 3,19,11)( 4,13,16)( 5, 7,21)( 8,20,10)( 9,14,15)(12,22,24) (17,23,18)$ $5, 5, 5, 5, 5$ $12$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $5, 5, 5, 5, 5$ $12$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

## Group invariants

 Order: $300=2^{2} \cdot 3 \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [300, 24]
 Character table:  2 2 2 2 2 2 2 2 2 2 2 2 2 . . 3 1 1 1 1 1 1 1 1 1 1 1 1 . . 5 2 . . . . . . . . . . . 2 2 1a 4a 4b 2a 3a 6a 12a 12b 12c 12d 6b 3b 5a 5b 2P 1a 2a 2a 1a 3b 3a 6a 6b 6a 6b 3b 3a 5a 5b 3P 1a 4b 4a 2a 1a 2a 4b 4a 4a 4b 2a 1a 5a 5b 5P 1a 4a 4b 2a 3b 6b 12d 12c 12b 12a 6a 3a 1a 1a 7P 1a 4b 4a 2a 3a 6a 12c 12d 12a 12b 6b 3b 5a 5b 11P 1a 4b 4a 2a 3b 6b 12b 12a 12d 12c 6a 3a 5a 5b 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 B /B -B -/B -B -/B B /B 1 1 X.4 1 -1 -1 1 /B B -/B -B -/B -B /B B 1 1 X.5 1 1 1 1 B /B B /B B /B B /B 1 1 X.6 1 1 1 1 /B B /B B /B B /B B 1 1 X.7 1 A -A -1 1 -1 A -A -A A -1 1 1 1 X.8 1 -A A -1 1 -1 -A A A -A -1 1 1 1 X.9 1 A -A -1 B -/B C /C -C -/C -B /B 1 1 X.10 1 A -A -1 /B -B -/C -C /C C -/B B 1 1 X.11 1 -A A -1 B -/B -C -/C C /C -B /B 1 1 X.12 1 -A A -1 /B -B /C C -/C -C -/B B 1 1 X.13 12 . . . . . . . . . . . 2 -3 X.14 12 . . . . . . . . . . . -3 2 A = -E(4) = -Sqrt(-1) = -i B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 C = -E(12)^11