Properties

Label 20T34
Order \(120\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $34$
Group :  $C_5\times S_4$
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)(17,18,19,20), (1,9,17,5,13)(2,10,18,6,14)(3,12,19,8,15,4,11,20,7,16)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
6:  $S_3$
10:  $C_{10}$
24:  $S_4$
30:  $S_3 \times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 5: $C_5$

Degree 10: None

Low degree siblings

30T33, 30T34, 40T64

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)(13,14,15,16)(17,18,19,20)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$
$ 10, 5, 5 $ $6$ $10$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 8,11,16,19, 4, 7,12,15,20)$
$ 15, 5 $ $8$ $15$ $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$
$ 10, 10 $ $3$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$
$ 20 $ $6$ $20$ $( 1, 6,11,16,17, 2, 7,12,13,18, 3, 8, 9,14,19, 4, 5,10,15,20)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
$ 10, 5, 5 $ $6$ $10$ $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,12,19, 8,15, 4,11,20, 7,16)$
$ 15, 5 $ $8$ $15$ $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$
$ 10, 10 $ $3$ $10$ $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$
$ 20 $ $6$ $20$ $( 1,10,19, 8,13, 2,11,20, 5,14, 3,12,17, 6,15, 4, 9,18, 7,16)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1,13, 5,17, 9)( 2,14, 6,18,10)( 3,15, 7,19,11)( 4,16, 8,20,12)$
$ 10, 5, 5 $ $6$ $10$ $( 1,13, 5,17, 9)( 2,14, 6,18,10)( 3,16, 7,20,11, 4,15, 8,19,12)$
$ 15, 5 $ $8$ $15$ $( 1,13, 5,17, 9)( 2,15, 8,18,11, 4,14, 7,20,10, 3,16, 6,19,12)$
$ 10, 10 $ $3$ $10$ $( 1,14, 5,18, 9, 2,13, 6,17,10)( 3,16, 7,20,11, 4,15, 8,19,12)$
$ 20 $ $6$ $20$ $( 1,14, 7,20, 9, 2,15, 8,17,10, 3,16, 5,18,11, 4,13, 6,19,12)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1,17,13, 9, 5)( 2,18,14,10, 6)( 3,19,15,11, 7)( 4,20,16,12, 8)$
$ 10, 5, 5 $ $6$ $10$ $( 1,17,13, 9, 5)( 2,18,14,10, 6)( 3,20,15,12, 7, 4,19,16,11, 8)$
$ 15, 5 $ $8$ $15$ $( 1,17,13, 9, 5)( 2,19,16,10, 7, 4,18,15,12, 6, 3,20,14,11, 8)$
$ 10, 10 $ $3$ $10$ $( 1,18,13,10, 5, 2,17,14, 9, 6)( 3,20,15,12, 7, 4,19,16,11, 8)$
$ 20 $ $6$ $20$ $( 1,18,15,12, 5, 2,19,16, 9, 6, 3,20,13,10, 7, 4,17,14,11, 8)$

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 37]
Character table: Data not available.