# Properties

 Label 16T60 Degree $16$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $\SL(2,3):C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(16, 60);

## Group action invariants

 Degree $n$: $16$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $60$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\SL(2,3):C_2$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $4$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,14,5,11,16,8,2,13,6,12,15,7)(3,10,4,9), (1,13)(2,14)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $A_4$

Degree 8: $A_4\times C_2$

## Low degree siblings

24T21

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 1, 1, 1, 1$ $4$ $3$ $( 3, 6,15)( 4, 5,16)( 7,14, 9)( 8,13,10)$ $3, 3, 3, 3, 1, 1, 1, 1$ $4$ $3$ $( 3,15, 6)( 4,16, 5)( 7, 9,14)( 8,10,13)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $6, 6, 2, 2$ $4$ $6$ $( 1, 2)( 3, 5,15, 4, 6,16)( 7,13, 9, 8,14,10)(11,12)$ $6, 6, 2, 2$ $4$ $6$ $( 1, 2)( 3,16, 6, 4,15, 5)( 7,10,14, 8, 9,13)(11,12)$ $4, 4, 4, 4$ $6$ $4$ $( 1, 3, 2, 4)( 5,16, 6,15)( 7,14, 8,13)( 9,12,10,11)$ $12, 4$ $4$ $12$ $( 1, 7,15,12, 6,13, 2, 8,16,11, 5,14)( 3, 9, 4,10)$ $12, 4$ $4$ $12$ $( 1, 7, 4,11, 5,10, 2, 8, 3,12, 6, 9)(13,16,14,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,11)( 6,12)( 9,15)(10,16)$ $12, 4$ $4$ $12$ $( 1, 8,15,11, 6,14, 2, 7,16,12, 5,13)( 3,10, 4, 9)$ $12, 4$ $4$ $12$ $( 1, 8, 4,12, 5, 9, 2, 7, 3,11, 6,10)(13,15,14,16)$ $4, 4, 4, 4$ $1$ $4$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 8, 6, 7)(13,15,14,16)$ $4, 4, 4, 4$ $1$ $4$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $48=2^{4} \cdot 3$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 48.33 magma: IdentifyGroup(G);
 Character table:  2 4 2 2 4 2 2 3 2 2 3 2 2 4 4 3 1 1 1 1 1 1 . 1 1 . 1 1 1 1 1a 3a 3b 2a 6a 6b 4a 12a 12b 2b 12c 12d 4b 4c 2P 1a 3b 3a 1a 3b 3a 2a 6a 6b 1a 6a 6b 2a 2a 3P 1a 1a 1a 2a 2a 2a 4a 4c 4b 2b 4b 4c 4c 4b 5P 1a 3b 3a 2a 6b 6a 4a 12d 12c 2b 12b 12a 4b 4c 7P 1a 3a 3b 2a 6a 6b 4a 12c 12d 2b 12a 12b 4c 4b 11P 1a 3b 3a 2a 6b 6a 4a 12b 12a 2b 12d 12c 4c 4b 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 A /A 1 A /A 1 -/A -A -1 -/A -A -1 -1 X.4 1 /A A 1 /A A 1 -A -/A -1 -A -/A -1 -1 X.5 1 A /A 1 A /A 1 /A A 1 /A A 1 1 X.6 1 /A A 1 /A A 1 A /A 1 A /A 1 1 X.7 2 -1 -1 -2 1 1 . B -B . -B B D -D X.8 2 -1 -1 -2 1 1 . -B B . B -B -D D X.9 2 -/A -A -2 /A A . C /C . -C -/C D -D X.10 2 -/A -A -2 /A A . -C -/C . C /C -D D X.11 2 -A -/A -2 A /A . -/C -C . /C C D -D X.12 2 -A -/A -2 A /A . /C C . -/C -C -D D X.13 3 . . 3 . . -1 . . -1 . . 3 3 X.14 3 . . 3 . . -1 . . 1 . . -3 -3 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -E(4) = -Sqrt(-1) = -i C = -E(12)^11 D = 2*E(4) = 2*Sqrt(-1) = 2i 

magma: CharacterTable(G);