Properties

Label 15T38
Order \(1500\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $38$
CHM label :  $[5^{3}:4]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5 $ $12$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $5$ $( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)$
$ 5, 5, 5 $ $12$ $5$ $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 1,13,10, 7, 4)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 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, 2, 2, 2, 2, 2, 1, 1, 1 $ $125$ $2$ $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$
$ 6, 6, 3 $ $125$ $6$ $( 1, 6, 5, 4, 3, 8)( 2, 7,15,11,13, 9)(10,12,14)$
$ 6, 6, 3 $ $125$ $6$ $( 1,11, 3,13,14,15)( 2,12, 4, 8, 6,10)( 5, 9, 7)$
$ 4, 4, 4, 1, 1, 1 $ $125$ $4$ $( 4, 7,13,10)( 5, 8,14,11)( 6, 9,15,12)$
$ 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, 4, 4, 1, 1, 1 $ $125$ $4$ $( 4,10,13, 7)( 5,11,14, 8)( 6,12,15, 9)$
$ 12, 3 $ $125$ $12$ $( 1, 6, 2, 7, 9,11, 4,15,14,13,12, 5)( 3, 8,10)$
$ 12, 3 $ $125$ $12$ $( 1,11, 9)( 2,12,10, 8,15, 4, 5, 6, 7,14, 3,13)$

Group invariants

Order:  $1500=2^{2} \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1500, 119]
Character table: Data not available.