Group action invariants
| Degree $n$ : | $37$ | |
| Transitive number $t$ : | $4$ | |
| Group : | $C_{37}:C_{4}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (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), (1,31,36,6)(2,25,35,12)(3,19,34,18)(4,13,33,24)(5,7,32,30)(8,26,29,11)(9,20,28,17)(10,14,27,23)(15,21,22,16) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 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 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $37$ | $4$ | $( 2, 7,37,32)( 3,13,36,26)( 4,19,35,20)( 5,25,34,14)( 6,31,33, 8)( 9,12,30,27) (10,18,29,21)(11,24,28,15)(16,17,23,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $37$ | $4$ | $( 2,32,37, 7)( 3,26,36,13)( 4,20,35,19)( 5,14,34,25)( 6, 8,33,31)( 9,27,30,12) (10,21,29,18)(11,15,28,24)(16,22,23,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $37$ | $2$ | $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27) (13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| $ 37 $ | $4$ | $37$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ |
| $ 37 $ | $4$ | $37$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ |
| $ 37 $ | $4$ | $37$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ |
| $ 37 $ | $4$ | $37$ | $( 1, 5, 9,13,17,21,25,29,33,37, 4, 8,12,16,20,24,28,32,36, 3, 7,11,15,19,23, 27,31,35, 2, 6,10,14,18,22,26,30,34)$ |
| $ 37 $ | $4$ | $37$ | $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$ |
| $ 37 $ | $4$ | $37$ | $( 1, 9,17,25,33, 4,12,20,28,36, 7,15,23,31, 2,10,18,26,34, 5,13,21,29,37, 8, 16,24,32, 3,11,19,27,35, 6,14,22,30)$ |
| $ 37 $ | $4$ | $37$ | $( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$ |
| $ 37 $ | $4$ | $37$ | $( 1,11,21,31, 4,14,24,34, 7,17,27,37,10,20,30, 3,13,23,33, 6,16,26,36, 9,19, 29, 2,12,22,32, 5,15,25,35, 8,18,28)$ |
| $ 37 $ | $4$ | $37$ | $( 1,16,31, 9,24, 2,17,32,10,25, 3,18,33,11,26, 4,19,34,12,27, 5,20,35,13,28, 6,21,36,14,29, 7,22,37,15,30, 8,23)$ |
Group invariants
| Order: | $148=2^{2} \cdot 37$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [148, 3] |
| Character table: |
2 2 2 2 2 . . . . . . . . .
37 1 . . . 1 1 1 1 1 1 1 1 1
1a 4a 4b 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i
2P 1a 2a 2a 1a 37b 37d 37a 37f 37h 37i 37c 37g 37e
3P 1a 4b 4a 2a 37c 37a 37g 37b 37i 37d 37h 37e 37f
5P 1a 4a 4b 2a 37e 37h 37i 37g 37b 37c 37f 37d 37a
7P 1a 4b 4a 2a 37e 37h 37i 37g 37b 37c 37f 37d 37a
11P 1a 4b 4a 2a 37f 37i 37d 37e 37c 37h 37b 37a 37g
13P 1a 4a 4b 2a 37d 37f 37b 37i 37g 37e 37a 37c 37h
17P 1a 4a 4b 2a 37g 37c 37h 37a 37f 37b 37e 37i 37d
19P 1a 4b 4a 2a 37c 37a 37g 37b 37i 37d 37h 37e 37f
23P 1a 4b 4a 2a 37h 37g 37e 37c 37d 37a 37i 37f 37b
29P 1a 4a 4b 2a 37f 37i 37d 37e 37c 37h 37b 37a 37g
31P 1a 4b 4a 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i
37P 1a 4a 4b 2a 1a 1a 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 -1 1 1 1 1 1 1 1 1 1
X.4 1 -A A -1 1 1 1 1 1 1 1 1 1
X.5 4 . . . B F D E I C H J G
X.6 4 . . . C G E I D J F B H
X.7 4 . . . D B H F G E J I C
X.8 4 . . . E C F G H I B D J
X.9 4 . . . F E B C J G D H I
X.10 4 . . . G I C J B H E F D
X.11 4 . . . H D J B C F I G E
X.12 4 . . . I J G H F D C E B
X.13 4 . . . J H I D E B G C F
A = -E(4)
= -Sqrt(-1) = -i
B = E(37)^9+E(37)^17+E(37)^20+E(37)^28
C = E(37)^2+E(37)^12+E(37)^25+E(37)^35
D = E(37)^10+E(37)^14+E(37)^23+E(37)^27
E = E(37)+E(37)^6+E(37)^31+E(37)^36
F = E(37)^3+E(37)^18+E(37)^19+E(37)^34
G = E(37)^4+E(37)^13+E(37)^24+E(37)^33
H = E(37)^5+E(37)^7+E(37)^30+E(37)^32
I = E(37)^8+E(37)^11+E(37)^26+E(37)^29
J = E(37)^15+E(37)^16+E(37)^21+E(37)^22
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