Properties

Label 30T41
Order \(180\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times S_3^2$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $41$
Group :  $C_5\times S_3^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14), (1,26,7,17,13,23,4,29,10,20)(2,27,8,18,14,24,5,30,11,21)(3,25,9,16,15,22,6,28,12,19)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
6:  $S_3$ x 2
10:  $C_{10}$ x 3
12:  $D_{6}$ x 2
20:  20T3
30:  $S_3 \times C_5$ x 2
36:  $S_3^2$
60:  30T12 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $C_5$

Degree 6: $S_3^2$

Degree 10: $C_{10}$

Degree 15: None

Low degree siblings

45T15

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 28]
Character table: Data not available.