Group action invariants
| Degree $n$ : | $43$ | |
| Transitive number $t$ : | $4$ | |
| Group : | $C_{43}:C_{6}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,37,36,42,6,7)(2,31,29,41,12,14)(3,25,22,40,18,21)(4,19,15,39,24,28)(5,13,8,38,30,35)(9,32,23,34,11,20)(10,26,16,33,17,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $43$ | $3$ | $( 2, 7,37)( 3,13,30)( 4,19,23)( 5,25,16)( 6,31, 9)( 8,43,38)(10,12,24) (11,18,17)(14,36,39)(15,42,32)(20,29,40)(21,35,33)(22,41,26)(27,28,34)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 1 $ | $43$ | $6$ | $( 2, 8, 7,43,37,38)( 3,15,13,42,30,32)( 4,22,19,41,23,26)( 5,29,25,40,16,20) ( 6,36,31,39, 9,14)(10,21,12,35,24,33)(11,28,18,34,17,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $43$ | $3$ | $( 2,37, 7)( 3,30,13)( 4,23,19)( 5,16,25)( 6, 9,31)( 8,38,43)(10,24,12) (11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)$ |
| $ 6, 6, 6, 6, 6, 6, 6, 1 $ | $43$ | $6$ | $( 2,38,37,43, 7, 8)( 3,32,30,42,13,15)( 4,26,23,41,19,22)( 5,20,16,40,25,29) ( 6,14, 9,39,31,36)(10,33,24,35,12,21)(11,27,17,34,18,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $43$ | $2$ | $( 2,43)( 3,42)( 4,41)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,34)(12,33) (13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$ |
| $ 43 $ | $6$ | $43$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$ |
| $ 43 $ | $6$ | $43$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42)$ |
| $ 43 $ | $6$ | $43$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40,43, 3, 6, 9,12,15,18,21,24,27,30, 33,36,39,42, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41)$ |
| $ 43 $ | $6$ | $43$ | $( 1, 5, 9,13,17,21,25,29,33,37,41, 2, 6,10,14,18,22,26,30,34,38,42, 3, 7,11, 15,19,23,27,31,35,39,43, 4, 8,12,16,20,24,28,32,36,40)$ |
| $ 43 $ | $6$ | $43$ | $( 1, 6,11,16,21,26,31,36,41, 3, 8,13,18,23,28,33,38,43, 5,10,15,20,25,30,35, 40, 2, 7,12,17,22,27,32,37,42, 4, 9,14,19,24,29,34,39)$ |
| $ 43 $ | $6$ | $43$ | $( 1,10,19,28,37, 3,12,21,30,39, 5,14,23,32,41, 7,16,25,34,43, 9,18,27,36, 2, 11,20,29,38, 4,13,22,31,40, 6,15,24,33,42, 8,17,26,35)$ |
| $ 43 $ | $6$ | $43$ | $( 1,11,21,31,41, 8,18,28,38, 5,15,25,35, 2,12,22,32,42, 9,19,29,39, 6,16,26, 36, 3,13,23,33,43,10,20,30,40, 7,17,27,37, 4,14,24,34)$ |
Group invariants
| Order: | $258=2 \cdot 3 \cdot 43$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [258, 1] |
| Character table: |
2 1 1 1 1 1 1 . . . . . . .
3 1 1 1 1 1 1 . . . . . . .
43 1 . . . . . 1 1 1 1 1 1 1
1a 3a 6a 3b 6b 2a 43a 43b 43c 43d 43e 43f 43g
2P 1a 3b 3a 3a 3b 1a 43b 43d 43a 43e 43g 43c 43f
3P 1a 1a 2a 1a 2a 2a 43c 43a 43f 43b 43d 43g 43e
5P 1a 3b 6b 3a 6a 2a 43e 43g 43d 43f 43c 43b 43a
7P 1a 3a 6a 3b 6b 2a 43a 43b 43c 43d 43e 43f 43g
11P 1a 3b 6b 3a 6a 2a 43f 43c 43g 43a 43b 43e 43d
13P 1a 3a 6a 3b 6b 2a 43e 43g 43d 43f 43c 43b 43a
17P 1a 3b 6b 3a 6a 2a 43g 43f 43e 43c 43a 43d 43b
19P 1a 3a 6a 3b 6b 2a 43d 43e 43b 43g 43f 43a 43c
23P 1a 3b 6b 3a 6a 2a 43f 43c 43g 43a 43b 43e 43d
29P 1a 3b 6b 3a 6a 2a 43b 43d 43a 43e 43g 43c 43f
31P 1a 3a 6a 3b 6b 2a 43b 43d 43a 43e 43g 43c 43f
37P 1a 3a 6a 3b 6b 2a 43a 43b 43c 43d 43e 43f 43g
41P 1a 3b 6b 3a 6a 2a 43b 43d 43a 43e 43g 43c 43f
43P 1a 3a 6a 3b 6b 2a 1a 1a 1a 1a 1a 1a 1a
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 1 A -/A /A -A -1 1 1 1 1 1 1 1
X.4 1 /A -A A -/A -1 1 1 1 1 1 1 1
X.5 1 A /A /A A 1 1 1 1 1 1 1 1
X.6 1 /A A A /A 1 1 1 1 1 1 1 1
X.7 6 . . . . . B F H E G C D
X.8 6 . . . . . C H D B F G E
X.9 6 . . . . . D C G H B E F
X.10 6 . . . . . E G F D C B H
X.11 6 . . . . . F E B G D H C
X.12 6 . . . . . G D E C H F B
X.13 6 . . . . . H B C F E D G
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(43)^9+E(43)^11+E(43)^20+E(43)^23+E(43)^32+E(43)^34
C = E(43)^5+E(43)^8+E(43)^13+E(43)^30+E(43)^35+E(43)^38
D = E(43)^4+E(43)^15+E(43)^19+E(43)^24+E(43)^28+E(43)^39
E = E(43)+E(43)^6+E(43)^7+E(43)^36+E(43)^37+E(43)^42
F = E(43)^3+E(43)^18+E(43)^21+E(43)^22+E(43)^25+E(43)^40
G = E(43)^2+E(43)^12+E(43)^14+E(43)^29+E(43)^31+E(43)^41
H = E(43)^10+E(43)^16+E(43)^17+E(43)^26+E(43)^27+E(43)^33
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