Properties

Label 30T15
Order \(90\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_{15}\times S_3$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $15$
Group :  $C_{15}\times S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29), (1,30,13,26,11,22,7,19,5,18,3,29,15,25,10,24,9,21,4,17,2,28,14,27,12,23,8,20,6,16)
$|\Aut(F/K)|$:  $15$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
5:  $C_5$
6:  $S_3$, $C_6$
10:  $C_{10}$
15:  $C_{15}$
18:  $S_3\times C_3$
30:  $S_3 \times C_5$, $C_{30}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $C_5$

Degree 6: $S_3\times C_3$

Degree 10: $C_{10}$

Degree 15: None

Low degree siblings

45T3

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1, 4, 7,12,15)( 2, 5, 8,10,13)( 3, 6, 9,11,14)(16,19,24,27,30,18,21,23,26, 29,17,20,22,25,28)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1, 4, 7,12,15)( 2, 5, 8,10,13)( 3, 6, 9,11,14)(16,20,23,27,28,17,21,24,25, 29,18,19,22,26,30)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 7,12,15)( 2, 5, 8,10,13)( 3, 6, 9,11,14)(16,21,22,27,29) (17,19,23,25,30)(18,20,24,26,28)$
$ 15, 15 $ $1$ $15$ $( 1, 5, 9,12,13, 3, 4, 8,11,15, 2, 6, 7,10,14)(16,19,24,27,30,18,21,23,26,29, 17,20,22,25,28)$
$ 15, 15 $ $2$ $15$ $( 1, 5, 9,12,13, 3, 4, 8,11,15, 2, 6, 7,10,14)(16,20,23,27,28,17,21,24,25,29, 18,19,22,26,30)$
$ 15, 15 $ $1$ $15$ $( 1, 6, 8,12,14, 2, 4, 9,10,15, 3, 5, 7,11,13)(16,20,23,27,28,17,21,24,25,29, 18,19,22,26,30)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 7,15, 4,12)( 2, 8,13, 5,10)( 3, 9,14, 6,11)(16,22,29,21,27) (17,23,30,19,25)(18,24,28,20,26)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1, 7,15, 4,12)( 2, 8,13, 5,10)( 3, 9,14, 6,11)(16,23,28,21,25,18,22,30,20, 27,17,24,29,19,26)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1, 7,15, 4,12)( 2, 8,13, 5,10)( 3, 9,14, 6,11)(16,24,30,21,26,17,22,28,19, 27,18,23,29,20,25)$
$ 15, 15 $ $1$ $15$ $( 1, 8,14, 4,10, 3, 7,13, 6,12, 2, 9,15, 5,11)(16,23,28,21,25,18,22,30,20,27, 17,24,29,19,26)$
$ 15, 15 $ $2$ $15$ $( 1, 8,14, 4,10, 3, 7,13, 6,12, 2, 9,15, 5,11)(16,24,30,21,26,17,22,28,19,27, 18,23,29,20,25)$
$ 15, 15 $ $1$ $15$ $( 1, 9,13, 4,11, 2, 7,14, 5,12, 3, 8,15, 6,10)(16,24,30,21,26,17,22,28,19,27, 18,23,29,20,25)$
$ 15, 15 $ $1$ $15$ $( 1,10, 6,15, 8, 3,12, 5,14, 7, 2,11, 4,13, 9)(16,25,20,29,23,18,27,19,28,22, 17,26,21,30,24)$
$ 15, 15 $ $2$ $15$ $( 1,10, 6,15, 8, 3,12, 5,14, 7, 2,11, 4,13, 9)(16,26,19,29,24,17,27,20,30,22, 18,25,21,28,23)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1,10, 6,15, 8, 3,12, 5,14, 7, 2,11, 4,13, 9)(16,27,21,29,22)(17,25,19,30,23) (18,26,20,28,24)$
$ 15, 15 $ $1$ $15$ $( 1,11, 5,15, 9, 2,12, 6,13, 7, 3,10, 4,14, 8)(16,26,19,29,24,17,27,20,30,22, 18,25,21,28,23)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1,11, 5,15, 9, 2,12, 6,13, 7, 3,10, 4,14, 8)(16,27,21,29,22)(17,25,19,30,23) (18,26,20,28,24)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,12, 4,15, 7)( 2,10, 5,13, 8)( 3,11, 6,14, 9)(16,27,21,29,22) (17,25,19,30,23)(18,26,20,28,24)$
$ 15, 15 $ $2$ $15$ $( 1,13,11, 7, 5, 3,15,10, 9, 4, 2,14,12, 8, 6)(16,28,25,22,20,17,29,26,23,21, 18,30,27,24,19)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1,13,11, 7, 5, 3,15,10, 9, 4, 2,14,12, 8, 6)(16,29,27,22,21)(17,30,25,23,19) (18,28,26,24,20)$
$ 15, 15 $ $1$ $15$ $( 1,13,11, 7, 5, 3,15,10, 9, 4, 2,14,12, 8, 6)(16,30,26,22,19,18,29,25,24,21, 17,28,27,23,20)$
$ 15, 15 $ $1$ $15$ $( 1,14,10, 7, 6, 2,15,11, 8, 4, 3,13,12, 9, 5)(16,28,25,22,20,17,29,26,23,21, 18,30,27,24,19)$
$ 15, 5, 5, 5 $ $2$ $15$ $( 1,14,10, 7, 6, 2,15,11, 8, 4, 3,13,12, 9, 5)(16,29,27,22,21)(17,30,25,23,19) (18,28,26,24,20)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,15,12, 7, 4)( 2,13,10, 8, 5)( 3,14,11, 9, 6)(16,29,27,22,21) (17,30,25,23,19)(18,28,26,24,20)$
$ 10, 10, 10 $ $3$ $10$ $( 1,16, 4,21, 7,22,12,27,15,29)( 2,17, 5,19, 8,23,10,25,13,30)( 3,18, 6,20, 9, 24,11,26,14,28)$
$ 30 $ $3$ $30$ $( 1,16, 5,19, 9,24,12,27,13,30, 3,18, 4,21, 8,23,11,26,15,29, 2,17, 6,20, 7, 22,10,25,14,28)$
$ 30 $ $3$ $30$ $( 1,16, 6,20, 8,23,12,27,14,28, 2,17, 4,21, 9,24,10,25,15,29, 3,18, 5,19, 7, 22,11,26,13,30)$
$ 30 $ $3$ $30$ $( 1,19,10,28, 6,22,15,17, 8,26, 3,21,12,30, 5,24,14,16, 7,25, 2,20,11,29, 4, 23,13,18, 9,27)$
$ 30 $ $3$ $30$ $( 1,19,11,29, 5,24,15,17, 9,27, 2,20,12,30, 6,22,13,18, 7,25, 3,21,10,28, 4, 23,14,16, 8,26)$
$ 10, 10, 10 $ $3$ $10$ $( 1,19,12,30, 4,23,15,17, 7,25)( 2,20,10,28, 5,24,13,18, 8,26)( 3,21,11,29, 6, 22,14,16, 9,27)$
$ 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,22, 2,23, 3,24)( 4,27, 5,25, 6,26)( 7,29, 8,30, 9,28)(10,17,11,18,12,16) (13,19,14,20,15,21)$
$ 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,22, 3,24, 2,23)( 4,27, 6,26, 5,25)( 7,29, 9,28, 8,30)(10,17,12,16,11,18) (13,19,15,21,14,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1,22)( 2,23)( 3,24)( 4,27)( 5,25)( 6,26)( 7,29)( 8,30)( 9,28)(10,17)(11,18) (12,16)(13,19)(14,20)(15,21)$
$ 30 $ $3$ $30$ $( 1,25, 9,16,13,24, 4,30,11,21, 2,26, 7,17,14,22, 5,28,12,19, 3,27, 8,18,15, 23, 6,29,10,20)$
$ 10, 10, 10 $ $3$ $10$ $( 1,25, 7,17,15,23, 4,30,12,19)( 2,26, 8,18,13,24, 5,28,10,20)( 3,27, 9,16,14, 22, 6,29,11,21)$
$ 30 $ $3$ $30$ $( 1,25, 8,18,14,22, 4,30,10,20, 3,27, 7,17,13,24, 6,29,12,19, 2,26, 9,16,15, 23, 5,28,11,21)$
$ 30 $ $3$ $30$ $( 1,28,14,25,10,22, 7,20, 6,17, 2,29,15,26,11,23, 8,21, 4,18, 3,30,13,27,12, 24, 9,19, 5,16)$
$ 10, 10, 10 $ $3$ $10$ $( 1,28,15,26,12,24, 7,20, 4,18)( 2,29,13,27,10,22, 8,21, 5,16)( 3,30,14,25,11, 23, 9,19, 6,17)$
$ 30 $ $3$ $30$ $( 1,28,13,27,11,23, 7,20, 5,16, 3,30,15,26,10,22, 9,19, 4,18, 2,29,14,25,12, 24, 8,21, 6,17)$

Group invariants

Order:  $90=2 \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [90, 6]
Character table: Data not available.