Group action invariants
| Degree $n$ : | $38$ | |
| Transitive number $t$ : | $6$ | |
| Group : | $C_2\times D_{19}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,11,6,2,12,5)(3,25,28,4,26,27)(7,15,33,8,16,34)(9,29,18,10,30,17)(13,20,24,14,19,23)(21,38,35,22,37,36)(31,32), (1,3)(2,4)(5,37)(6,38)(7,35)(8,36)(9,33)(10,34)(11,31)(12,32)(13,29)(14,30)(15,28)(16,27)(17,25)(18,26)(19,23)(20,24)(21,22) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $C_6$ x 3 12: $C_6\times C_2$ 114: $C_{19}:C_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: $C_{19}:C_{6}$
Low degree siblings
38T6Siblings 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, 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, 1, 1 $ | $19$ | $3$ | $( 3,16,23)( 4,15,24)( 5,29, 7)( 6,30, 8)( 9,19,13)(10,20,14)(11,33,36) (12,34,35)(17,38,26)(18,37,25)(21,27,31)(22,28,32)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $19$ | $6$ | $( 3,17,16,38,23,26)( 4,18,15,37,24,25)( 5,33,29,36, 7,11)( 6,34,30,35, 8,12) ( 9,27,19,31,13,21)(10,28,20,32,14,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $19$ | $3$ | $( 3,23,16)( 4,24,15)( 5, 7,29)( 6, 8,30)( 9,13,19)(10,14,20)(11,36,33) (12,35,34)(17,26,38)(18,25,37)(21,31,27)(22,32,28)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $19$ | $6$ | $( 3,26,23,38,16,17)( 4,25,24,37,15,18)( 5,11, 7,36,29,33)( 6,12, 8,35,30,34) ( 9,21,13,31,19,27)(10,22,14,32,20,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $19$ | $2$ | $( 3,38)( 4,37)( 5,36)( 6,35)( 7,33)( 8,34)( 9,31)(10,32)(11,29)(12,30)(13,27) (14,28)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1, 2)( 3,15,23, 4,16,24)( 5,30, 7, 6,29, 8)( 9,20,13,10,19,14) (11,34,36,12,33,35)(17,37,26,18,38,25)(21,28,31,22,27,32)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1, 2)( 3,18,16,37,23,25)( 4,17,15,38,24,26)( 5,34,29,35, 7,12) ( 6,33,30,36, 8,11)( 9,28,19,32,13,22)(10,27,20,31,14,21)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1, 2)( 3,24,16, 4,23,15)( 5, 8,29, 6, 7,30)( 9,14,19,10,13,20) (11,35,33,12,36,34)(17,25,38,18,26,37)(21,32,27,22,31,28)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $19$ | $6$ | $( 1, 2)( 3,25,23,37,16,18)( 4,26,24,38,15,17)( 5,12, 7,35,29,34) ( 6,11, 8,36,30,33)( 9,22,13,32,19,28)(10,21,14,31,20,27)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $19$ | $2$ | $( 1, 2)( 3,37)( 4,38)( 5,35)( 6,36)( 7,34)( 8,33)( 9,32)(10,31)(11,30)(12,29) (13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| $ 38 $ | $6$ | $38$ | $( 1, 3, 6, 7, 9,11,13,16,18,20,21,23,25,28,30,32,34,36,37, 2, 4, 5, 8,10,12, 14,15,17,19,22,24,26,27,29,31,33,35,38)$ |
| $ 19, 19 $ | $6$ | $19$ | $( 1, 4, 6, 8, 9,12,13,15,18,19,21,24,25,27,30,31,34,35,37)( 2, 3, 5, 7,10,11, 14,16,17,20,22,23,26,28,29,32,33,36,38)$ |
| $ 38 $ | $6$ | $38$ | $( 1, 5, 9,14,18,22,25,29,34,38, 4, 7,12,16,19,23,27,32,35, 2, 6,10,13,17,21, 26,30,33,37, 3, 8,11,15,20,24,28,31,36)$ |
| $ 19, 19 $ | $6$ | $19$ | $( 1, 6, 9,13,18,21,25,30,34,37, 4, 8,12,15,19,24,27,31,35)( 2, 5,10,14,17,22, 26,29,33,38, 3, 7,11,16,20,23,28,32,36)$ |
| $ 19, 19 $ | $6$ | $19$ | $( 1, 9,18,25,34, 4,12,19,27,35, 6,13,21,30,37, 8,15,24,31)( 2,10,17,26,33, 3, 11,20,28,36, 5,14,22,29,38, 7,16,23,32)$ |
| $ 38 $ | $6$ | $38$ | $( 1,10,18,26,34, 3,12,20,27,36, 6,14,21,29,37, 7,15,23,31, 2, 9,17,25,33, 4, 11,19,28,35, 5,13,22,30,38, 8,16,24,32)$ |
Group invariants
| Order: | $228=2^{2} \cdot 3 \cdot 19$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [228, 7] |
| Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1
3 1 1 1 1 1 1 1 1 1 1 1 1 . . . . . .
19 1 . . . . . 1 . . . . . 1 1 1 1 1 1
1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c
2P 1a 3b 3a 3a 3b 1a 1a 3b 3a 3a 3b 1a 19b 19b 19c 19c 19a 19a
3P 1a 1a 2a 1a 2a 2a 2b 2b 2c 2b 2c 2c 38b 19b 38c 19c 19a 38a
5P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38b 19b 38c 19c 19a 38a
7P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c
11P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38a 19a 38b 19b 19c 38c
13P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38c 19c 38a 19a 19b 38b
17P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38b 19b 38c 19c 19a 38a
19P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 2b 1a 2b 1a 1a 2b
23P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38c 19c 38a 19a 19b 38b
29P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38c 19c 38a 19a 19b 38b
31P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c
37P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c
X.1 1 1 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 1 1 -1
X.3 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 1
X.4 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 -1
X.5 1 A -/A /A -A -1 -1 -A /A -/A A 1 -1 1 -1 1 1 -1
X.6 1 /A -A A -/A -1 -1 -/A A -A /A 1 -1 1 -1 1 1 -1
X.7 1 A -/A /A -A -1 1 A -/A /A -A -1 1 1 1 1 1 1
X.8 1 /A -A A -/A -1 1 /A -A A -/A -1 1 1 1 1 1 1
X.9 1 A /A /A A 1 -1 -A -/A -/A -A -1 -1 1 -1 1 1 -1
X.10 1 /A A A /A 1 -1 -/A -A -A -/A -1 -1 1 -1 1 1 -1
X.11 1 A /A /A A 1 1 A /A /A A 1 1 1 1 1 1 1
X.12 1 /A A A /A 1 1 /A A A /A 1 1 1 1 1 1 1
X.13 6 . . . . . 6 . . . . . B B C C D D
X.14 6 . . . . . 6 . . . . . C C D D B B
X.15 6 . . . . . 6 . . . . . D D B B C C
X.16 6 . . . . . -6 . . . . . -B B -C C D -D
X.17 6 . . . . . -6 . . . . . -C C -D D B -B
X.18 6 . . . . . -6 . . . . . -D D -B B C -C
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17
C = E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15
D = E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18
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