Properties

 Label 30T31 Order $$120$$ n $$30$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:S_4$

Group action invariants

 Degree $n$ : $30$ Transitive number $t$ : $31$ Group : $C_5:S_4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,23,2,24)(3,22,4,21)(5,29,6,30)(7,28,8,27)(9,26,10,25)(13,19)(14,20)(15,18)(16,17), (1,11,24,5,16,27,9,19,21,3,13,25,7,17,30)(2,12,23,6,15,28,10,20,22,4,14,26,8,18,29) $|\Aut(F/K)|$: $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
10:  $D_{5}$
24:  $S_4$
30:  $D_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $S_4$

Degree 10: None

Degree 15: $D_{15}$

Low degree siblings

20T33, 30T19, 40T63

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [120, 38]
 Character table:  2 3 3 2 2 2 2 2 2 . . . . . 3 1 . . . 1 . . 1 1 1 1 1 1 5 1 1 . . 1 1 1 1 1 1 1 1 1 1a 2a 4a 2b 5a 10a 10b 5b 15a 3a 15b 15c 15d 2P 1a 1a 2a 1a 5b 5b 5a 5a 15c 3a 15d 15b 15a 3P 1a 2a 4a 2b 5b 10b 10a 5a 5b 1a 5b 5a 5a 5P 1a 2a 4a 2b 1a 2a 2a 1a 3a 3a 3a 3a 3a 7P 1a 2a 4a 2b 5b 10b 10a 5a 15d 3a 15c 15a 15b 11P 1a 2a 4a 2b 5a 10a 10b 5b 15b 3a 15a 15d 15c 13P 1a 2a 4a 2b 5b 10b 10a 5a 15c 3a 15d 15b 15a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 X.3 2 2 . . 2 2 2 2 -1 -1 -1 -1 -1 X.4 2 2 . . A A *A *A A 2 A *A *A X.5 2 2 . . *A *A A A *A 2 *A A A X.6 2 2 . . *A *A A A C -1 D E F X.7 2 2 . . *A *A A A D -1 C F E X.8 2 2 . . A A *A *A E -1 F D C X.9 2 2 . . A A *A *A F -1 E C D X.10 3 -1 -1 1 3 -1 -1 3 . . . . . X.11 3 -1 1 -1 3 -1 -1 3 . . . . . X.12 6 -2 . . B -A -*A *B . . . . . X.13 6 -2 . . *B -*A -A B . . . . . A = E(5)+E(5)^4 = (-1+Sqrt(5))/2 = b5 B = 3*E(5)+3*E(5)^4 = (-3+3*Sqrt(5))/2 = 3b5 C = E(15)^4+E(15)^11 D = E(15)+E(15)^14 E = E(15)^7+E(15)^8 F = E(15)^2+E(15)^13