# Properties

 Label 15T8 Degree $15$ Order $60$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_5\times C_3$

# Learn more

Show commands: Magma

magma: G := TransitiveGroup(15, 8);

## Group action invariants

 Degree $n$: $15$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $8$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $F_5\times C_3$ CHM label: $F(5)[x]3$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $3$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,7,4,13)(2,14,8,11)(3,6,12,9), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 5: $F_5$

## Low degree siblings

30T7

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $60=2^{2} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 60.6 magma: IdentifyGroup(G);
 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 1 1 1 5 1 . . . 1 . . . . 1 . . 1 1 1 1a 2a 4a 4b 15a 6a 12a 12b 12c 15b 6b 12d 5a 3a 3b 2P 1a 1a 2a 2a 15b 3a 6b 6b 6a 15a 3b 6a 5a 3b 3a 3P 1a 2a 4b 4a 5a 2a 4b 4a 4a 5a 2a 4b 5a 1a 1a 5P 1a 2a 4a 4b 3a 6b 12d 12c 12b 3b 6a 12a 1a 3b 3a 7P 1a 2a 4b 4a 15a 6a 12b 12a 12d 15b 6b 12c 5a 3a 3b 11P 1a 2a 4b 4a 15b 6b 12c 12d 12a 15a 6a 12b 5a 3b 3a 13P 1a 2a 4a 4b 15a 6a 12a 12b 12c 15b 6b 12d 5a 3a 3b X.1 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 1 X.3 1 -1 A -A 1 -1 A -A -A 1 -1 A 1 1 1 X.4 1 -1 -A A 1 -1 -A A A 1 -1 -A 1 1 1 X.5 1 -1 A -A B -B C -C /C /B -/B -/C 1 /B B X.6 1 -1 A -A /B -/B -/C /C -C B -B C 1 B /B X.7 1 -1 -A A B -B -C C -/C /B -/B /C 1 /B B X.8 1 -1 -A A /B -/B /C -/C C B -B -C 1 B /B X.9 1 1 -1 -1 B B -B -B -/B /B /B -/B 1 /B B X.10 1 1 -1 -1 /B /B -/B -/B -B B B -B 1 B /B X.11 1 1 1 1 B B B B /B /B /B /B 1 /B B X.12 1 1 1 1 /B /B /B /B B B B B 1 B /B X.13 4 . . . -1 . . . . -1 . . -1 4 4 X.14 4 . . . -/B . . . . -B . . -1 D /D X.15 4 . . . -B . . . . -/B . . -1 /D D A = -E(4) = -Sqrt(-1) = -i B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 C = -E(12)^11 D = 4*E(3)^2 = -2-2*Sqrt(-3) = -2-2i3 

magma: CharacterTable(G);