Group action invariants
| Degree $n$ : | $45$ | |
| Transitive number $t$ : | $45$ | |
| Group : | $C_3:S_5$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,40,11,33)(2,42,10,31)(3,41,12,32)(4,8,23,36)(5,9,24,35)(6,7,22,34)(13,43)(14,45)(15,44)(16,38,27,19)(17,37,26,21)(18,39,25,20)(28,30), (1,43,26,35)(2,45,27,36)(3,44,25,34)(4,41,17,28)(5,42,18,30)(6,40,16,29)(7,19)(8,20)(9,21)(10,37,22,14)(11,39,23,15)(12,38,24,13)(32,33) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 120: $S_5$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 3: $S_3$
Degree 5: $S_5$
Degree 9: None
Degree 15: $S_5$, $\GL(2,4):C_2$ x 2, $\GL(2,4):C_2$
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 $ | $15$ | $2$ | $( 4,14)( 5,13)( 6,15)( 7,38)( 8,37)( 9,39)(10,28)(11,29)(12,30)(19,42)(20,41) (21,40)(25,44)(26,45)(27,43)(31,34)(32,35)(33,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1 $ | $90$ | $4$ | $( 2, 3)( 4,21,14,40)( 5,20,13,41)( 6,19,15,42)( 7,28,38,10)( 8,29,37,11) ( 9,30,39,12)(16,24)(17,23)(18,22)(25,32,44,35)(26,33,45,36)(27,31,43,34)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $30$ | $2$ | $( 2, 3)( 4,26)( 5,27)( 6,25)( 7,39)( 8,37)( 9,38)(10,12)(13,43)(14,45)(15,44) (16,24)(17,23)(18,22)(19,35)(20,34)(21,36)(28,30)(31,41)(32,42)(33,40)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,44,45)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $30$ | $6$ | $( 1, 2, 3)( 4,15, 5,14, 6,13)( 7,37, 9,38, 8,39)(10,30,11,28,12,29)(16,18,17) (19,40,20,42,21,41)(22,24,23)(25,45,27,44,26,43)(31,36,32,34,33,35)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 3 $ | $60$ | $6$ | $( 1, 4,23,11,17,26)( 2, 5,22,12,16,25)( 3, 6,24,10,18,27)( 7,39,44,20,34,15) ( 8,37,45,21,36,14)( 9,38,43,19,35,13)(28,31,41,30,32,42)(29,33,40)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 4,21,40,14)( 2, 6,20,41,15)( 3, 5,19,42,13)( 7,38,44,24,25) ( 8,37,45,23,26)( 9,39,43,22,27)(10,32,35,28,16)(11,33,36,29,17) (12,31,34,30,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $20$ | $3$ | $( 1, 4,36)( 2, 6,35)( 3, 5,34)( 7,31,19)( 8,33,21)( 9,32,20)(10,39,16) (11,37,17)(12,38,18)(13,30,44)(14,29,45)(15,28,43)(22,27,41)(23,26,40) (24,25,42)$ |
| $ 15, 15, 15 $ | $24$ | $15$ | $( 1, 5,20,40,13, 2, 4,19,41,14, 3, 6,21,42,15)( 7,39,45,24,27, 8,38,43,23,25, 9,37,44,22,26)(10,33,34,28,17,12,32,36,30,16,11,31,35,29,18)$ |
| $ 15, 15, 15 $ | $24$ | $15$ | $( 1, 5,28,37,31, 2, 4,30,39,33, 3, 6,29,38,32)( 7,16,11,44,41, 8,18,10,45,42, 9,17,12,43,40)(13,35,21,24,27,14,34,20,23,25,15,36,19,22,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $40$ | $3$ | $( 1, 5,35)( 2, 4,34)( 3, 6,36)( 7,32,21)( 8,31,20)( 9,33,19)(10,37,18) (11,38,16)(12,39,17)(13,28,45)(14,30,43)(15,29,44)(22,26,42)(23,25,41) (24,27,40)$ |
Group invariants
| Order: | $360=2^{3} \cdot 3^{2} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | [360, 120] |
| Character table: |
2 3 3 2 2 2 2 1 . 1 . . .
3 2 1 . 1 2 1 1 1 2 1 1 2
5 1 . . . 1 . . 1 . 1 1 .
1a 2a 4a 2b 3a 6a 6b 5a 3b 15a 15b 3c
2P 1a 1a 2a 1a 3a 3a 3b 5a 3b 15a 15b 3c
3P 1a 2a 4a 2b 1a 2a 2b 5a 1a 5a 5a 1a
5P 1a 2a 4a 2b 3a 6a 6b 1a 3b 3a 3a 3c
7P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c
11P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c
13P 1a 2a 4a 2b 3a 6a 6b 5a 3b 15b 15a 3c
X.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
X.3 2 2 . . -1 -1 . 2 2 -1 -1 -1
X.4 4 . . -2 4 . 1 -1 1 -1 -1 1
X.5 4 . . 2 4 . -1 -1 1 -1 -1 1
X.6 5 1 -1 1 5 1 1 . -1 . . -1
X.7 5 1 1 -1 5 1 -1 . -1 . . -1
X.8 6 -2 . . 6 -2 . 1 . 1 1 .
X.9 6 -2 . . -3 1 . 1 . A /A .
X.10 6 -2 . . -3 1 . 1 . /A A .
X.11 8 . . . -4 . . -2 2 1 1 -1
X.12 10 2 . . -5 -1 . . -2 . . 1
A = -E(15)-E(15)^2-E(15)^4-E(15)^8
= (-1-Sqrt(-15))/2 = -1-b15
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