Properties

Label 30T42
Order \(180\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_{15}$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $42$
Group :  $S_3\times D_{15}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29), (2,3)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(16,28)(17,30)(18,29)(19,27)(20,26)(21,25)(22,23), (1,17,6,19,7,23,12,25,13,28)(2,18,4,20,8,24,10,26,14,29)(3,16,5,21,9,22,11,27,15,30)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
10:  $D_{5}$
12:  $D_{6}$ x 2
20:  $D_{10}$
30:  $D_{15}$
36:  $S_3^2$
60:  $D_5\times S_3$, $D_{30}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $S_3^2$

Degree 10: $D_{10}$

Degree 15: None

Low degree siblings

45T13

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 $ $4$ $3$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 2, 3)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(16,28)(17,30)(18,29)(19,27) (20,26)(21,25)(22,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
$ 15, 15 $ $2$ $15$ $( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
$ 15, 15 $ $4$ $15$ $( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
$ 15, 15 $ $2$ $15$ $( 1, 5, 8,12,15, 2, 6, 9,10,13, 3, 4, 7,11,14)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 5, 8,12,15, 2, 6, 9,10,13, 3, 4, 7,11,14)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 6, 7,12,13)( 2, 4, 8,10,14)( 3, 5, 9,11,15)(16,21,22,27,30) (17,19,23,25,28)(18,20,24,26,29)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,22,30,21,27) (17,23,28,19,25)(18,24,29,20,26)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,23,29,21,25,18,22,28,20, 27,17,24,30,19,26)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,24,28,21,26,17,22,29,19, 27,18,23,30,20,25)$
$ 15, 15 $ $2$ $15$ $( 1, 8,15, 6,10, 3, 7,14, 5,12, 2, 9,13, 4,11)(16,23,29,21,25,18,22,28,20,27, 17,24,30,19,26)$
$ 15, 15 $ $4$ $15$ $( 1, 8,15, 6,10, 3, 7,14, 5,12, 2, 9,13, 4,11)(16,24,28,21,26,17,22,29,19,27, 18,23,30,20,25)$
$ 15, 15 $ $2$ $15$ $( 1, 9,14, 6,11, 2, 7,15, 4,12, 3, 8,13, 5,10)(16,24,28,21,26,17,22,29,19,27, 18,23,30,20,25)$
$ 30 $ $6$ $30$ $( 1,16, 4,19, 9,24,12,27,14,28, 3,18, 6,21, 8,23,11,26,13,30, 2,17, 5,20, 7, 22,10,25,15,29)$
$ 30 $ $6$ $30$ $( 1,16, 5,20, 8,23,12,27,15,29, 2,17, 6,21, 9,24,10,25,13,30, 3,18, 4,19, 7, 22,11,26,14,28)$
$ 10, 10, 10 $ $6$ $10$ $( 1,16, 6,21, 7,22,12,27,13,30)( 2,17, 4,19, 8,23,10,25,14,28)( 3,18, 5,20, 9, 24,11,26,15,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1,16)( 2,18)( 3,17)( 4,29)( 5,28)( 6,30)( 7,27)( 8,26)( 9,25)(10,24)(11,23) (12,22)(13,21)(14,20)(15,19)$
$ 6, 6, 6, 6, 6 $ $30$ $6$ $( 1,16, 2,18, 3,17)( 4,29, 5,28, 6,30)( 7,27, 8,26, 9,25)(10,24,11,23,12,22) (13,21,14,20,15,19)$
$ 30 $ $6$ $30$ $( 1,19,10,29, 5,22,13,17, 8,26, 3,21,12,28, 4,24,15,16, 7,25, 2,20,11,30, 6, 23,14,18, 9,27)$
$ 30 $ $6$ $30$ $( 1,19,11,30, 4,24,13,17, 9,27, 2,20,12,28, 5,22,14,18, 7,25, 3,21,10,29, 6, 23,15,16, 8,26)$
$ 10, 10, 10 $ $6$ $10$ $( 1,19,12,28, 6,23,13,17, 7,25)( 2,20,10,29, 4,24,14,18, 8,26)( 3,21,11,30, 5, 22,15,16, 9,27)$
$ 6, 6, 6, 6, 6 $ $6$ $6$ $( 1,22, 2,23, 3,24)( 4,25, 5,26, 6,27)( 7,30, 8,28, 9,29)(10,17,11,18,12,16) (13,21,14,19,15,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,30)( 8,28)( 9,29)(10,17)(11,18) (12,16)(13,21)(14,19)(15,20)$

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 29]
Character table: Data not available.