Group action invariants
| Degree $n$ : | $27$ | |
| Transitive number $t$ : | $8$ | |
| Group : | $D_{27}$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,22)(2,24)(3,23)(4,21)(5,20)(6,19)(7,17)(8,16)(9,18)(10,15)(11,14)(12,13)(25,26), (1,18,6,19,9,22,12,26,14,2,16,4,20,7,23,10,27,15,3,17,5,21,8,24,11,25,13) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 18: $D_{9}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 9: $D_{9}$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $27$ | $2$ | $( 2, 3)( 4,27)( 5,26)( 6,25)( 7,23)( 8,22)( 9,24)(10,20)(11,19)(12,21)(13,18) (14,17)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
| $ 27 $ | $2$ | $27$ | $( 1, 4, 8,12,15,18,20,24,26, 3, 6, 7,11,14,17,19,23,25, 2, 5, 9,10,13,16,21, 22,27)$ |
| $ 27 $ | $2$ | $27$ | $( 1, 5, 7,12,13,17,20,22,25, 3, 4, 9,11,15,16,19,24,27, 2, 6, 8,10,14,18,21, 23,26)$ |
| $ 27 $ | $2$ | $27$ | $( 1, 6, 9,12,14,16,20,23,27, 3, 5, 8,11,13,18,19,22,26, 2, 4, 7,10,15,17,21, 24,25)$ |
| $ 27 $ | $2$ | $27$ | $( 1, 7,13,20,25, 4,11,16,24, 2, 8,14,21,26, 5,12,17,22, 3, 9,15,19,27, 6,10, 18,23)$ |
| $ 27 $ | $2$ | $27$ | $( 1, 8,15,20,26, 6,11,17,23, 2, 9,13,21,27, 4,12,18,24, 3, 7,14,19,25, 5,10, 16,22)$ |
| $ 27 $ | $2$ | $27$ | $( 1, 9,14,20,27, 5,11,18,22, 2, 7,15,21,25, 6,12,16,23, 3, 8,13,19,26, 4,10, 17,24)$ |
| $ 9, 9, 9 $ | $2$ | $9$ | $( 1,10,19, 3,12,21, 2,11,20)( 4,13,23, 6,15,22, 5,14,24)( 7,18,27, 9,17,26, 8, 16,25)$ |
| $ 9, 9, 9 $ | $2$ | $9$ | $( 1,11,21, 3,10,20, 2,12,19)( 4,14,22, 6,13,24, 5,15,23)( 7,16,26, 9,18,25, 8, 17,27)$ |
| $ 9, 9, 9 $ | $2$ | $9$ | $( 1,12,20, 3,11,19, 2,10,21)( 4,15,24, 6,14,23, 5,13,22)( 7,17,25, 9,16,27, 8, 18,26)$ |
| $ 27 $ | $2$ | $27$ | $( 1,13,25,11,24, 8,21, 5,17, 3,15,27,10,23, 7,20, 4,16, 2,14,26,12,22, 9,19, 6,18)$ |
| $ 27 $ | $2$ | $27$ | $( 1,14,27,11,22, 7,21, 6,16, 3,13,26,10,24, 9,20, 5,18, 2,15,25,12,23, 8,19, 4,17)$ |
| $ 27 $ | $2$ | $27$ | $( 1,15,26,11,23, 9,21, 4,18, 3,14,25,10,22, 8,20, 6,17, 2,13,27,12,24, 7,19, 5,16)$ |
Group invariants
| Order: | $54=2 \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [54, 1] |
| Character table: |
2 1 1 . . . . . . . . . . . . .
3 3 . 3 3 3 3 3 3 3 3 3 3 3 3 3
1a 2a 3a 27a 27b 27c 27d 27e 27f 9a 9b 9c 27g 27h 27i
2P 1a 1a 3a 27e 27d 27f 27g 27i 27h 9b 9c 9a 27c 27a 27b
3P 1a 2a 1a 9c 9c 9c 9a 9a 9a 3a 3a 3a 9b 9b 9b
5P 1a 2a 3a 27g 27h 27i 27a 27c 27b 9c 9a 9b 27e 27d 27f
7P 1a 2a 3a 27f 27e 27d 27i 27h 27g 9b 9c 9a 27b 27c 27a
11P 1a 2a 3a 27d 27f 27e 27h 27g 27i 9b 9c 9a 27a 27b 27c
13P 1a 2a 3a 27h 27i 27g 27b 27a 27c 9c 9a 9b 27d 27f 27e
17P 1a 2a 3a 27c 27a 27b 27e 27f 27d 9a 9b 9c 27i 27g 27h
19P 1a 2a 3a 27b 27c 27a 27f 27d 27e 9a 9b 9c 27h 27i 27g
23P 1a 2a 3a 27i 27g 27h 27c 27b 27a 9c 9a 9b 27f 27e 27d
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 -1 -1 -1 -1 2 2 2 -1 -1 -1
X.4 2 . 2 A A A C C C -1 -1 -1 B B B
X.5 2 . 2 B B B A A A -1 -1 -1 C C C
X.6 2 . 2 C C C B B B -1 -1 -1 A A A
X.7 2 . -1 D F E L J K A C B G I H
X.8 2 . -1 E D F J K L A C B H G I
X.9 2 . -1 F E D K L J A C B I H G
X.10 2 . -1 G I H D E F B A C J L K
X.11 2 . -1 H G I E F D B A C K J L
X.12 2 . -1 I H G F D E B A C L K J
X.13 2 . -1 J L K G H I C B A E D F
X.14 2 . -1 K J L H I G C B A F E D
X.15 2 . -1 L K J I G H C B A D F E
A = -E(9)^2-E(9)^4-E(9)^5-E(9)^7
B = E(9)^4+E(9)^5
C = E(9)^2+E(9)^7
D = E(27)^13+E(27)^14
E = E(27)^5+E(27)^22
F = -E(27)^5-E(27)^13-E(27)^14-E(27)^22
G = E(27)^11+E(27)^16
H = -E(27)^7-E(27)^11-E(27)^16-E(27)^20
I = E(27)^7+E(27)^20
J = -E(27)^8-E(27)^10-E(27)^17-E(27)^19
K = E(27)^10+E(27)^17
L = E(27)^8+E(27)^19
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