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Properties

Label 30T22
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $22$
Group :  $S_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,5,2,8,6)(3,9,11,17,21,20)(4,10,12,18,22,19)(13,26,23,30,27,16)(14,25,24,29,28,15), (1,29,26,4)(2,30,25,3)(5,22,23,14)(6,21,24,13)(7,27,12,16)(8,28,11,15)(9,17,10,18)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 5: $S_5$

Degree 6: $\PGL(2,5)$

Degree 10: None

Degree 15: $S_5$

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T25, 30T27, 40T62

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [120, 34]
Character table:    
     2  3  3  2  2  .  1  1
     3  1  .  .  1  .  1  1
     5  1  .  .  .  1  .  .

       1a 2a 4a 2b 5a 6a 3a
    2P 1a 1a 2a 1a 5a 3a 3a
    3P 1a 2a 4a 2b 5a 2b 1a
    5P 1a 2a 4a 2b 1a 6a 3a

X.1     1  1  1  1  1  1  1
X.2     1  1 -1 -1  1 -1  1
X.3     4  .  . -2 -1  1  1
X.4     4  .  .  2 -1 -1  1
X.5     5  1 -1  1  .  1 -1
X.6     5  1  1 -1  . -1 -1
X.7     6 -2  .  .  1  .  .