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Properties

Label 15T31
Order \(750\)
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 $t$ :  $31$
CHM label :  $1/2[5^{3}:2]S(3)$
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8), (3,6,9,12,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
10:  $D_{5}$
30:  $D_{15}$
150:  $(C_5^2 : C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T31 x 3, 30T186 x 4, 30T187 x 2

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 $ $6$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $6$ $5$ $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,12, 6,15, 9)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 1,13,10, 7, 4)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 1,10, 4,13, 7)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 1, 4, 7,10,13)( 3,15,12, 9, 6)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)$
$ 5, 5, 5 $ $6$ $5$ $( 1,10, 4,13, 7)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 1,13,10, 7, 4)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $2$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $6$ $5$ $( 1,10, 4,13, 7)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 15 $ $50$ $15$ $( 1, 6,14, 4, 9, 2, 7,12, 5,10,15, 8,13, 3,11)$
$ 15 $ $50$ $15$ $( 1,11, 6, 4,14, 9, 7, 2,12,10, 5,15,13, 8, 3)$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $6$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 1,10, 4,13, 7)( 2,14,11, 8, 5)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1,10, 4,13, 7)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 1,13,10, 7, 4)( 2,11, 5,14, 8)$
$ 5, 5, 5 $ $2$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 15 $ $50$ $15$ $( 1, 6, 2, 7,12, 8,13, 3,14, 4, 9, 5,10,15,11)$
$ 15 $ $50$ $15$ $( 1,11, 6, 7, 2,12,13, 8, 3, 4,14, 9,10, 5,15)$
$ 10, 2, 2, 1 $ $75$ $10$ $( 2,15,11, 6, 5,12,14, 3, 8, 9)( 4,13)( 7,10)$
$ 10, 2, 2, 1 $ $75$ $10$ $( 1,13)( 2,12, 5, 9, 8, 6,11, 3,14,15)( 4,10)$
$ 10, 2, 2, 1 $ $75$ $10$ $( 1,10)( 2, 9,14,12,11,15, 8, 3, 5, 6)( 4, 7)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $75$ $2$ $( 1, 4)( 2, 3)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)$
$ 10, 2, 2, 1 $ $75$ $10$ $( 1, 7)( 2, 6, 8,15,14, 9, 5, 3,11,12)(10,13)$

Group invariants

Order:  $750=2 \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [750, 27]
Character table: Data not available.