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Properties

Label 30T51
Order \(240\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\times F_5$

Related objects

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $51$
Group :  $C_2\times S_3\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,15,9,27)(4,16,10,28)(5,30,17,23)(6,29,18,24)(7,13,25,20)(8,14,26,19)(11,12)(21,22), (1,15,24,19,21,5,14,10,11,25,4,29)(2,16,23,20,22,6,13,9,12,26,3,30)(7,27,17)(8,28,18), (3,18,9,6)(4,17,10,5)(7,19,25,14)(8,20,26,13)(11,21)(12,22)(15,24,27,29)(16,23,28,30)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
6:  $S_3$
8:  $C_4\times C_2$ x 6, $C_2^3$
12:  $D_{6}$ x 3
16:  $C_4\times C_2^2$
20:  $F_5$
24:  $S_3 \times C_2^2$, $S_3 \times C_4$ x 2
40:  $F_{5}\times C_2$ x 3
48:  24T27
80:  20T16
120:  $F_5 \times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $D_{6}$

Degree 10: $F_{5}\times C_2$

Degree 15: $F_5 \times S_3$

Low degree siblings

30T51 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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1 $ $15$ $4$ $( 3, 6, 9,18)( 4, 5,10,17)( 7,14,25,19)( 8,13,26,20)(11,21)(12,22) (15,29,27,24)(16,30,28,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 9)( 4,10)( 5,17)( 6,18)( 7,25)( 8,26)(13,20)(14,19)(15,27)(16,28)(23,30) (24,29)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $5$ $4$ $( 3,16, 9,28)( 4,15,10,27)( 5,29,17,24)( 6,30,18,23)( 7,14,25,19)( 8,13,26,20)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1 $ $15$ $4$ $( 3,18, 9, 6)( 4,17,10, 5)( 7,19,25,14)( 8,20,26,13)(11,21)(12,22) (15,24,27,29)(16,23,28,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3,23)( 4,24)( 5,15)( 6,16)( 9,30)(10,29)(11,21)(12,22)(17,27)(18,28)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $5$ $4$ $( 3,28, 9,16)( 4,27,10,15)( 5,24,17,29)( 6,23,18,30)( 7,19,25,14)( 8,20,26,13)$
$ 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $15$ $4$ $( 1, 2)( 3, 5, 9,17)( 4, 6,10,18)( 7,13,25,20)( 8,14,26,19)(11,22)(12,21) (15,30,27,23)(16,29,28,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5,18)( 6,17)( 7,26)( 8,25)(11,12)(13,19)(14,20)(15,28) (16,27)(21,22)(23,29)(24,30)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $5$ $4$ $( 1, 2)( 3,15, 9,27)( 4,16,10,28)( 5,30,17,23)( 6,29,18,24)( 7,13,25,20) ( 8,14,26,19)(11,12)(21,22)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $15$ $4$ $( 1, 2)( 3,17, 9, 5)( 4,18,10, 6)( 7,20,25,13)( 8,19,26,14)(11,22)(12,21) (15,23,27,30)(16,24,28,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3,24)( 4,23)( 5,16)( 6,15)( 7, 8)( 9,29)(10,30)(11,22)(12,21)(13,14) (17,28)(18,27)(19,20)(25,26)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $5$ $4$ $( 1, 2)( 3,27, 9,15)( 4,28,10,16)( 5,23,17,30)( 6,24,18,29)( 7,20,25,13) ( 8,19,26,14)(11,12)(21,22)$
$ 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 $ $8$ $30$ $( 1, 3, 5, 8,10,12,14,16,17,20,21,23,25,28,29, 2, 4, 6, 7, 9,11,13,15,18,19, 22,24,26,27,30)$
$ 6, 6, 6, 6, 6 $ $10$ $6$ $( 1, 3,11,13,21,23)( 2, 4,12,14,22,24)( 5,20,15,30,25, 9)( 6,19,16,29,26,10) ( 7,28,17, 8,27,18)$
$ 12, 12, 6 $ $10$ $12$ $( 1, 3,17,26,21,23, 7,16,11,13,27, 6)( 2, 4,18,25,22,24, 8,15,12,14,28, 5) ( 9,29,20,10,30,19)$
$ 10, 10, 10 $ $12$ $10$ $( 1, 3,25,28,19,22,14,16, 7, 9)( 2, 4,26,27,20,21,13,15, 8,10)( 5,18,29,12,24, 6,17,30,11,23)$
$ 12, 12, 6 $ $10$ $12$ $( 1, 3,29, 8,21,23,19,28,11,13,10,18)( 2, 4,30, 7,22,24,20,27,12,14, 9,17) ( 5,26,15, 6,25,16)$
$ 15, 15 $ $8$ $15$ $( 1, 4, 5, 7,10,11,14,15,17,19,21,24,25,27,29)( 2, 3, 6, 8, 9,12,13,16,18,20, 22,23,26,28,30)$
$ 6, 6, 6, 6, 3, 3 $ $10$ $6$ $( 1, 4,11,14,21,24)( 2, 3,12,13,22,23)( 5,19,15,29,25,10)( 6,20,16,30,26, 9) ( 7,27,17)( 8,28,18)$
$ 12, 12, 3, 3 $ $10$ $12$ $( 1, 4,17,25,21,24, 7,15,11,14,27, 5)( 2, 3,18,26,22,23, 8,16,12,13,28, 6) ( 9,30,20)(10,29,19)$
$ 10, 10, 5, 5 $ $12$ $10$ $( 1, 4,25,27,19,21,14,15, 7,10)( 2, 3,26,28,20,22,13,16, 8, 9)( 5,17,29,11,24) ( 6,18,30,12,23)$
$ 12, 12, 3, 3 $ $10$ $12$ $( 1, 4,29, 7,21,24,19,27,11,14,10,17)( 2, 3,30, 8,22,23,20,28,12,13, 9,18) ( 5,25,15)( 6,26,16)$
$ 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,16,22,28)( 4,10,15,21,27) ( 5,11,17,24,29)( 6,12,18,23,30)$
$ 10, 10, 10 $ $4$ $10$ $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$
$ 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 $ $2$ $6$ $( 1,12,21, 2,11,22)( 3,14,23, 4,13,24)( 5,16,25, 6,15,26)( 7,18,27, 8,17,28) ( 9,19,30,10,20,29)$

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [240, 195]
Character table: Data not available.