Properties

Label 30T21
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\times D_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $21$
Group :  $C_2\times S_3\times D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23), (1,6)(2,5)(3,4)(7,30)(8,29)(9,27)(10,28)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19), (1,11)(2,12)(5,25)(6,26)(7,17)(8,18)(13,23)(14,24)(19,29)(20,30)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
10:  $D_{5}$
12:  $D_{6}$ x 3
20:  $D_{10}$ x 3
24:  $S_3 \times C_2^2$
40:  20T8
60:  $D_5\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $D_{6}$

Degree 10: $D_{10}$

Degree 15: $D_5\times S_3$

Low degree siblings

30T21 x 3

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 42]
Character table: Data not available.