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Properties

Label 15T38
Degree $15$
Order $1500$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^3:C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(15, 38);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^3:C_{12}$
CHM label:   $[5^{3}:4]3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (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), (3,6,9,12,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$
$60$:  $F_5\times C_3$
$300$:  $(C_5^2 : C_4):C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T38 x 7, 30T287 x 8

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{15}$ $1$ $1$ $()$
$5^{3}$ $12$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$5^{2},1^{5}$ $12$ $5$ $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$3^{5}$ $25$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$3^{5}$ $25$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$5,1^{10}$ $12$ $5$ $( 3, 6, 9,12,15)$
$5^{2},1^{5}$ $12$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)$
$5^{3}$ $12$ $5$ $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$5^{3}$ $12$ $5$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$5^{2},1^{5}$ $12$ $5$ $( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$5^{2},1^{5}$ $12$ $5$ $( 1,13,10, 7, 4)( 3, 9,15, 6,12)$
$5^{3}$ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$5^{3}$ $4$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$5^{3}$ $12$ $5$ $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$15$ $100$ $15$ $( 1, 6,14, 4, 9, 2, 7,12, 5,10,15, 8,13, 3,11)$
$15$ $100$ $15$ $( 1,11, 6, 4,14, 9, 7, 2,12,10, 5,15,13, 8, 3)$
$2^{6},1^{3}$ $125$ $2$ $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$
$6^{2},3$ $125$ $6$ $( 1, 6, 5, 4, 3, 8)( 2, 7,15,11,13, 9)(10,12,14)$
$6^{2},3$ $125$ $6$ $( 1,11, 3,13,14,15)( 2,12, 4, 8, 6,10)( 5, 9, 7)$
$4^{3},1^{3}$ $125$ $4$ $( 4, 7,13,10)( 5, 8,14,11)( 6, 9,15,12)$
$12,3$ $125$ $12$ $( 1, 6,14)( 2, 7, 3, 8, 4,12,11,10, 9, 5,13,15)$
$12,3$ $125$ $12$ $( 1,11,15, 7, 8, 9,10,14, 6, 4, 2,12)( 3,13, 5)$
$4^{3},1^{3}$ $125$ $4$ $( 4,10,13, 7)( 5,11,14, 8)( 6,12,15, 9)$
$12,3$ $125$ $12$ $( 1, 6, 2, 7, 9,11, 4,15,14,13,12, 5)( 3, 8,10)$
$12,3$ $125$ $12$ $( 1,11, 9)( 2,12,10, 8,15, 4, 5, 6, 7,14, 3,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1500=2^{2} \cdot 3 \cdot 5^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1500.119
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A1 4A-1 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 6A1 6A-1 12A1 12A-1 12A5 12A-5 15A1 15A-1
Size 1 125 25 25 125 125 4 12 12 12 12 12 12 12 12 12 12 125 125 125 125 125 125 100 100
2 P 1A 1A 3A-1 3A1 2A 2A 5A 5G 5H 5E 5D 5I 5K 5C 5J 5F 5B 3A1 3A-1 6A-1 6A1 6A-1 6A1 15A-1 15A1
3 P 1A 2A 1A 1A 4A-1 4A1 5A 5G 5H 5E 5D 5I 5K 5C 5J 5F 5B 2A 2A 4A-1 4A-1 4A1 4A1 5A 5A
5 P 1A 2A 3A-1 3A1 4A1 4A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 6A-1 6A1 12A-5 12A-1 12A1 12A5 3A-1 3A1
Type
1500.119.1a R 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
1500.119.1b R 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
1500.119.1c1 C 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
1500.119.1c2 C 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
1500.119.1d1 C 1 −1 1 1 −i i 1 1 1 1 1 1 1 1 1 1 1 −1 −1 i −i i −i 1 1
1500.119.1d2 C 1 −1 1 1 i −i 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −i i −i i 1 1
1500.119.1e1 C 1 1 ζ3−1 ζ3 −1 −1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
1500.119.1e2 C 1 1 ζ3 ζ3−1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
1500.119.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 1 1 1 1 1 1 1 1 1 1 1 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125 −ζ122 ζ124
1500.119.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 1 1 1 1 1 1 1 1 1 1 1 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12 ζ124 −ζ122
1500.119.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 1 1 1 1 1 1 1 1 1 1 1 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125 −ζ122 ζ124
1500.119.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 1 1 1 1 1 1 1 1 1 1 1 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12 ζ124 −ζ122
1500.119.4a R 4 0 4 4 0 0 −1 −1 −1 4 −1 4 −1 −1 −1 −1 −1 0 0 0 0 0 0 −1 −1
1500.119.4b1 C 4 0 4ζ3−1 4ζ3 0 0 −1 −1 −1 4 −1 4 −1 −1 −1 −1 −1 0 0 0 0 0 0 −ζ3−1 −ζ3
1500.119.4b2 C 4 0 4ζ3 4ζ3−1 0 0 −1 −1 −1 4 −1 4 −1 −1 −1 −1 −1 0 0 0 0 0 0 −ζ3 −ζ3−1
1500.119.12a R 12 0 0 0 0 0 12 −3 2 2 2 −3 −3 2 −3 −3 2 0 0 0 0 0 0 0 0
1500.119.12b R 12 0 0 0 0 0 12 2 −3 −3 −3 2 2 −3 2 2 −3 0 0 0 0 0 0 0 0
1500.119.12c R 12 0 0 0 0 0 −3 −3 −3 −3 2 2 −3 −3 2 2 7 0 0 0 0 0 0 0 0
1500.119.12d R 12 0 0 0 0 0 −3 −3 −3 2 2 −3 2 2 7 −3 −3 0 0 0 0 0 0 0 0
1500.119.12e R 12 0 0 0 0 0 −3 −3 2 2 −3 −3 7 −3 −3 2 2 0 0 0 0 0 0 0 0
1500.119.12f R 12 0 0 0 0 0 −3 −3 7 −3 −3 2 −3 2 2 2 −3 0 0 0 0 0 0 0 0
1500.119.12g R 12 0 0 0 0 0 −3 2 −3 −3 −3 2 2 7 −3 −3 2 0 0 0 0 0 0 0 0
1500.119.12h R 12 0 0 0 0 0 −3 2 −3 2 2 −3 −3 2 −3 7 −3 0 0 0 0 0 0 0 0
1500.119.12i R 12 0 0 0 0 0 −3 2 2 −3 7 2 2 −3 −3 −3 −3 0 0 0 0 0 0 0 0
1500.119.12j R 12 0 0 0 0 0 −3 7 2 2 −3 −3 −3 −3 2 −3 2 0 0 0 0 0 0 0 0

magma: CharacterTable(G);