Properties

 Label 20T33 Order $$120$$ n $$20$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:S_4$

Related objects

Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $33$ Group : $C_5:S_4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,9,17,5,13)(2,11,20,6,15,4,10,19,8,14,3,12,18,7,16), (1,7,4,6)(2,5,3,8)(9,19,12,18)(10,17,11,20)(13,15,16,14) $|\Aut(F/K)|$: $1$

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 4: $S_4$

Degree 5: $D_{5}$

Degree 10: None

Low degree siblings

30T19, 30T31, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $30$ $2$ $( 3, 4)( 5,17)( 6,18)( 7,20)( 8,19)( 9,13)(10,14)(11,16)(12,15)$ $3, 3, 3, 3, 3, 1, 1, 1, 1, 1$ $8$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ $4, 4, 4, 4, 4$ $30$ $4$ $( 1, 2, 3, 4)( 5,18, 7,20)( 6,19, 8,17)( 9,14,11,16)(10,15,12,13)$ $5, 5, 5, 5$ $2$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$ $15, 5$ $8$ $15$ $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$ $15, 5$ $8$ $15$ $( 1, 5, 9,13,17)( 2, 8,11,14,20, 3, 6,12,15,18, 4, 7,10,16,19)$ $10, 10$ $6$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$ $5, 5, 5, 5$ $2$ $5$ $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$ $15, 5$ $8$ $15$ $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$ $15, 5$ $8$ $15$ $( 1, 9,17, 5,13)( 2,12,19, 6,16, 3,10,20, 7,14, 4,11,18, 8,15)$ $10, 10$ $6$ $10$ $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$

Group invariants

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