# Properties

 Label 15T1 Order $$15$$ n $$15$$ Cyclic Yes Abelian Yes Solvable Yes Primitive No $p$-group No Group: $C_{15}$

# Related objects

## Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $1$ Group : $C_{15}$ CHM label : $C(15)=5[x]3$ Parity: $1$ Primitive: No Nilpotency class: $1$ Generators: (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) $|\Aut(F/K)|$: $15$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
5:  $C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 5: $C_5$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $15$ $1$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$ $15$ $1$ $15$ $( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$ $5, 5, 5$ $1$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ $15$ $1$ $15$ $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$ $3, 3, 3, 3, 3$ $1$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ $5, 5, 5$ $1$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ $15$ $1$ $15$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$ $15$ $1$ $15$ $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$ $5, 5, 5$ $1$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ $3, 3, 3, 3, 3$ $1$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ $15$ $1$ $15$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$ $5, 5, 5$ $1$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ $15$ $1$ $15$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$ $15$ $1$ $15$ $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

## Group invariants

 Order: $15=3 \cdot 5$ Cyclic: Yes Abelian: Yes Solvable: Yes GAP id: [15, 1]
 Character table:  3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 15a 15b 5a 15c 3a 5b 15d 15e 5c 3b 15f 5d 15g 15h X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 A /A 1 A /A 1 A /A 1 A /A X.3 1 /A A 1 /A A 1 /A A 1 /A A 1 /A A X.4 1 B E /E /B 1 B E /E /B 1 B E /E /B X.5 1 C G /E /D /A B F /F /B A D E /G /C X.6 1 D F /E /C A B G /G /B /A C E /F /D X.7 1 E /B B /E 1 E /B B /E 1 E /B B /E X.8 1 F /C B /G /A E /D D /E A G /B C /F X.9 1 G /D B /F A E /C C /E /A F /B D /G X.10 1 /E B /B E 1 /E B /B E 1 /E B /B E X.11 1 /G D /B F /A /E C /C E A /F B /D G X.12 1 /F C /B G A /E D /D E /A /G B /C F X.13 1 /B /E E B 1 /B /E E B 1 /B /E E B X.14 1 /D /F E C /A /B /G G B A /C /E F D X.15 1 /C /G E D A /B /F F B /A /D /E G C A = E(3) = (-1+Sqrt(-3))/2 = b3 B = E(5)^2 C = E(15)^11 D = E(15) E = E(5)^4 F = E(15)^2 G = E(15)^7