# Properties

 Label 15T19 Order $$300$$ n $$15$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $(C_5^2 : C_4):C_3$

# Related objects

## Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $19$ Group : $(C_5^2 : C_4):C_3$ CHM label : $[5^{2}:4]3$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9) $|\Aut(F/K)|$: $1$

## 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 $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 5: None

## Low degree siblings

15T19, 25T26, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1, 1$ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$ $4, 4, 4, 1, 1, 1$ $25$ $4$ $( 3, 9, 6,15)( 4, 7,13,10)( 5, 8,14,11)$ $4, 4, 4, 1, 1, 1$ $25$ $4$ $( 3,15, 6, 9)( 4,10,13, 7)( 5,11,14, 8)$ $5, 5, 1, 1, 1, 1, 1$ $12$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ $3, 3, 3, 3, 3$ $25$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$ $6, 6, 3$ $25$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$ $12, 3$ $25$ $12$ $( 1, 2, 3, 7,14,12,10, 5, 9, 4, 8,15)( 6,13,11)$ $12, 3$ $25$ $12$ $( 1, 2, 3,13, 8, 6, 7, 5,12,10,14, 9)( 4,11,15)$ $3, 3, 3, 3, 3$ $25$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$ $6, 6, 3$ $25$ $6$ $( 1, 3, 5,13, 6, 2)( 4,15, 8,10, 9,14)( 7,12,11)$ $12, 3$ $25$ $12$ $( 1, 3, 8,13,12,11, 4, 9, 5, 7,15, 2)( 6,14,10)$ $12, 3$ $25$ $12$ $( 1, 3,14, 7, 6, 8, 4,12,11,13, 9, 2)( 5,10,15)$ $5, 5, 5$ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$

## 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 2a 4a 4b 5a 3a 6a 12a 12b 3b 6b 12c 12d 5b 2P 1a 1a 2a 2a 5a 3b 3b 6b 6b 3a 3a 6a 6a 5b 3P 1a 2a 4b 4a 5a 1a 2a 4b 4a 1a 2a 4b 4a 5b 5P 1a 2a 4a 4b 1a 3b 6b 12c 12d 3a 6a 12a 12b 1a 7P 1a 2a 4b 4a 5a 3a 6a 12b 12a 3b 6b 12d 12c 5b 11P 1a 2a 4b 4a 5a 3b 6b 12d 12c 3a 6a 12b 12a 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 A -A 1 1 -1 A -A 1 -1 A -A 1 X.4 1 -1 -A A 1 1 -1 -A A 1 -1 -A A 1 X.5 1 -1 A -A 1 B -B C -C /B -/B -/C /C 1 X.6 1 -1 A -A 1 /B -/B -/C /C B -B C -C 1 X.7 1 -1 -A A 1 B -B -C C /B -/B /C -/C 1 X.8 1 -1 -A A 1 /B -/B /C -/C B -B -C C 1 X.9 1 1 -1 -1 1 B B -B -B /B /B -/B -/B 1 X.10 1 1 -1 -1 1 /B /B -/B -/B B B -B -B 1 X.11 1 1 1 1 1 B B B B /B /B /B /B 1 X.12 1 1 1 1 1 /B /B /B /B B B B B 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