Group action invariants
| Degree $n$: | $45$ | |
| Transitive number $t$: | $40$ | |
| Group: | $S_3\times A_5$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $3$ | |
| Generators: | (1,5,30,38,33,2,4,28,37,32)(3,6,29,39,31)(7,17,11,45,40,9,16,10,43,42)(8,18,12,44,41)(13,35,19,22,26,14,36,21,24,25)(15,34,20,23,27), (1,18,24,3,16,23)(2,17,22)(4,20,30,6,19,29)(5,21,28)(7,15,26,8,13,27)(9,14,25)(10,32,45)(11,31,43,12,33,44)(34,40,39,36,41,37)(35,42,38) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $60$: $A_5$ $120$: $A_5\times C_2$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 3: $S_3$
Degree 5: $A_5$
Degree 9: None
Degree 15: $A_5$, $A_5 \times S_3$
Low degree siblings
There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.
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, 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, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 4,13)( 5,14)( 6,15)( 7,37)( 8,39)( 9,38)(10,28)(11,30)(12,29)(19,40)(20,41) (21,42)(25,45)(26,43)(27,44)(31,34)(32,35)(33,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(10,12)(14,15)(17,18)(20,21)(22,23)(25,27)(28,29)(31,32) (34,35)(38,39)(41,42)(44,45)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $45$ | $2$ | $( 2, 3)( 4,13)( 5,15)( 6,14)( 7,37)( 8,38)( 9,39)(10,29)(11,30)(12,28)(17,18) (19,40)(20,42)(21,41)(22,23)(25,44)(26,43)(27,45)(31,35)(32,34)(33,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,42,41) (43,45,44)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $30$ | $6$ | $( 1, 2, 3)( 4,14, 6,13, 5,15)( 7,38, 8,37, 9,39)(10,29,11,28,12,30)(16,17,18) (19,42,20,40,21,41)(22,23,24)(25,44,26,45,27,43)(31,36,32,34,33,35)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 4,19,40,13)( 2, 5,21,42,14)( 3, 6,20,41,15)( 7,37,43,24,26) ( 8,39,44,23,27)( 9,38,45,22,25)(10,32,35,28,17)(11,33,36,30,16) (12,31,34,29,18)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 4,30,37,33)( 2, 5,28,38,32)( 3, 6,29,39,31)( 7,16,11,43,40) ( 8,18,12,44,41)( 9,17,10,45,42)(13,36,19,24,26)(14,35,21,22,25) (15,34,20,23,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $20$ | $3$ | $( 1, 4,36)( 2, 5,35)( 3, 6,34)( 7,33,19)( 8,31,20)( 9,32,21)(10,38,17) (11,37,16)(12,39,18)(13,30,43)(14,28,45)(15,29,44)(22,25,42)(23,27,41) (24,26,40)$ |
| $ 10, 10, 10, 5, 5, 5 $ | $36$ | $10$ | $( 1, 4,19,40,13)( 2, 6,21,41,14, 3, 5,20,42,15)( 7,37,43,24,26) ( 8,38,44,22,27, 9,39,45,23,25)(10,31,35,29,17,12,32,34,28,18)(11,33,36,30,16)$ |
| $ 10, 10, 10, 5, 5, 5 $ | $36$ | $10$ | $( 1, 4,30,37,33)( 2, 6,28,39,32, 3, 5,29,38,31)( 7,16,11,43,40) ( 8,17,12,45,41, 9,18,10,44,42)(13,36,19,24,26)(14,34,21,23,25,15,35,20,22,27)$ |
| $ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $ | $60$ | $6$ | $( 1, 4,36)( 2, 6,35, 3, 5,34)( 7,33,19)( 8,32,20, 9,31,21)(10,39,17,12,38,18) (11,37,16)(13,30,43)(14,29,45,15,28,44)(22,27,42,23,25,41)(24,26,40)$ |
| $ 15, 15, 15 $ | $24$ | $15$ | $( 1, 5,20,40,14, 3, 4,21,41,13, 2, 6,19,42,15)( 7,38,44,24,25, 8,37,45,23,26, 9,39,43,22,27)(10,31,36,28,18,11,32,34,30,17,12,33,35,29,16)$ |
| $ 15, 15, 15 $ | $24$ | $15$ | $( 1, 5,29,37,32, 3, 4,28,39,33, 2, 6,30,38,31)( 7,17,12,43,42, 8,16,10,44,40, 9,18,11,45,41)(13,35,20,24,25,15,36,21,23,26,14,34,19,22,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1, 5,34)( 2, 6,36)( 3, 4,35)( 7,32,20)( 8,33,21)( 9,31,19)(10,39,16) (11,38,18)(12,37,17)(13,28,44)(14,29,43)(15,30,45)(22,27,40)(23,26,42) (24,25,41)$ |
Group invariants
| Order: | $360=2^{3} \cdot 3^{2} \cdot 5$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | [360, 121] |
| Character table: |
2 3 3 3 3 2 2 1 1 1 1 1 1 . . .
3 2 1 1 . 2 1 1 1 2 . . 1 1 1 2
5 1 . 1 . 1 . 1 1 . 1 1 . 1 1 .
1a 2a 2b 2c 3a 6a 5a 5b 3b 10a 10b 6b 15a 15b 3c
2P 1a 1a 1a 1a 3a 3a 5b 5a 3b 5b 5a 3b 15b 15a 3c
3P 1a 2a 2b 2c 1a 2a 5b 5a 1a 10b 10a 2b 5b 5a 1a
5P 1a 2a 2b 2c 3a 6a 1a 1a 3b 2b 2b 6b 3a 3a 3c
7P 1a 2a 2b 2c 3a 6a 5b 5a 3b 10b 10a 6b 15b 15a 3c
11P 1a 2a 2b 2c 3a 6a 5a 5b 3b 10a 10b 6b 15a 15b 3c
13P 1a 2a 2b 2c 3a 6a 5b 5a 3b 10b 10a 6b 15b 15a 3c
X.1 1 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 1 1
X.3 2 2 . . -1 -1 2 2 2 . . . -1 -1 -1
X.4 3 -1 -3 1 3 -1 A *A . -A -*A . A *A .
X.5 3 -1 -3 1 3 -1 *A A . -*A -A . *A A .
X.6 3 -1 3 -1 3 -1 A *A . A *A . A *A .
X.7 3 -1 3 -1 3 -1 *A A . *A A . *A A .
X.8 4 . -4 . 4 . -1 -1 1 1 1 -1 -1 -1 1
X.9 4 . 4 . 4 . -1 -1 1 -1 -1 1 -1 -1 1
X.10 5 1 -5 -1 5 1 . . -1 . . 1 . . -1
X.11 5 1 5 1 5 1 . . -1 . . -1 . . -1
X.12 6 -2 . . -3 1 B *B . . . . -A -*A .
X.13 6 -2 . . -3 1 *B B . . . . -*A -A .
X.14 8 . . . -4 . -2 -2 2 . . . 1 1 -1
X.15 10 2 . . -5 -1 . . -2 . . . . . 1
A = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
B = -2*E(5)-2*E(5)^4
= 1-Sqrt(5) = 1-r5
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