↑

Properties

Label 15T18
Degree $15$
Order $300$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $((C_5^2 : C_3):C_2):C_2$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(15, 18);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $((C_5^2 : C_3):C_2):C_2$
CHM label:   $[5^{2}:2]S(3)$
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:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,13,10,7,4)(2,5,8,11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T18, 25T27, 30T66 x 2, 30T72 x 2, 30T80 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 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 $ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)$
$ 10, 2, 2, 1 $ $30$ $10$ $( 2, 3, 5,15, 8,12,11, 9,14, 6)( 4,13)( 7,10)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 10, 2, 2, 1 $ $30$ $10$ $( 2, 9, 8, 3,14,12, 5, 6,11,15)( 4,13)( 7,10)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $15$ $2$ $( 2,12)( 3,11)( 4,13)( 5, 9)( 6, 8)( 7,10)(14,15)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 6, 6, 3 $ $50$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$
$ 10, 5 $ $30$ $10$ $( 1, 2, 4, 5, 7, 8,10,11,13,14)( 3,15,12, 9, 6)$
$ 10, 5 $ $30$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $300=2^{2} \cdot 3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  300.25
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 5B1 5B2 6A 10A1 10A3 10B1 10B3
Size 1 15 15 25 50 6 6 6 6 50 30 30 30 30
2 P 1A 1A 1A 1A 3A 5B2 5B1 5A2 5A1 3A 5B1 5A1 5A2 5B2
3 P 1A 2A 2B 2C 1A 5B2 5B1 5A2 5A1 2C 10B3 10A3 10A1 10B1
5 P 1A 2A 2B 2C 3A 1A 1A 1A 1A 6A 2B 2A 2A 2B
Type
300.25.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
300.25.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
300.25.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 1 1
300.25.1d R 1 1 −1 −1 1 1 1 1 1 −1 1 1 −1 −1
300.25.2a R 2 0 0 2 −1 2 2 2 2 −1 0 0 0 0
300.25.2b R 2 0 0 −2 −1 2 2 2 2 1 0 0 0 0
300.25.6a1 R 6 0 2 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
300.25.6a2 R 6 0 2 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
300.25.6b1 R 6 2 0 0 0 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0
300.25.6b2 R 6 2 0 0 0 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0
300.25.6c1 R 6 −2 0 0 0 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
300.25.6c2 R 6 −2 0 0 0 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
300.25.6d1 R 6 0 −2 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
300.25.6d2 R 6 0 −2 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52

magma: CharacterTable(G);