Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $9$ | |
| Group : | $C_9:C_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,3,2,4)(5,34,6,33)(7,35,8,36)(9,32,10,31)(11,30,12,29)(13,28,14,27)(15,26,16,25)(17,23,18,24)(19,21,20,22), (1,34,32,28,24,19,16,12,7)(2,33,31,27,23,20,15,11,8)(3,35,29,25,21,17,14,10,6)(4,36,30,26,22,18,13,9,5) | |
| $|\Aut(F/K)|$: | $36$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 6: $S_3$ 18: $D_{9}$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $C_4$
Degree 6: $S_3$
Degree 9: $D_{9}$
Degree 12: $C_3 : C_4$
Degree 18: $D_9$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,35, 8,36)( 9,32,10,31)(11,30,12,29)(13,28,14,27) (15,26,16,25)(17,23,18,24)(19,21,20,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $9$ | $4$ | $( 1, 4, 2, 3)( 5,33, 6,34)( 7,36, 8,35)( 9,31,10,32)(11,29,12,30)(13,27,14,28) (15,25,16,26)(17,24,18,23)(19,22,20,21)$ |
| $ 9, 9, 9, 9 $ | $2$ | $9$ | $( 1, 7,12,16,19,24,28,32,34)( 2, 8,11,15,20,23,27,31,33)( 3, 6,10,14,17,21,25, 29,35)( 4, 5, 9,13,18,22,26,30,36)$ |
| $ 18, 18 $ | $2$ | $18$ | $( 1, 8,12,15,19,23,28,31,34, 2, 7,11,16,20,24,27,32,33)( 3, 5,10,13,17,22,25, 30,35, 4, 6, 9,14,18,21,26,29,36)$ |
| $ 18, 18 $ | $2$ | $18$ | $( 1,11,19,27,34, 8,16,23,32, 2,12,20,28,33, 7,15,24,31)( 3, 9,17,26,35, 5,14, 22,29, 4,10,18,25,36, 6,13,21,30)$ |
| $ 9, 9, 9, 9 $ | $2$ | $9$ | $( 1,12,19,28,34, 7,16,24,32)( 2,11,20,27,33, 8,15,23,31)( 3,10,17,25,35, 6,14, 21,29)( 4, 9,18,26,36, 5,13,22,30)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,28, 2,16,27)( 3,13,25, 4,14,26)( 5,17,30, 6,18,29)( 7,20,32, 8,19,31) ( 9,21,36,10,22,35)(11,24,33,12,23,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,28)( 2,15,27)( 3,14,25)( 4,13,26)( 5,18,30)( 6,17,29)( 7,19,32) ( 8,20,31)( 9,22,36)(10,21,35)(11,23,33)(12,24,34)$ |
| $ 9, 9, 9, 9 $ | $2$ | $9$ | $( 1,19,34,16,32,12,28, 7,24)( 2,20,33,15,31,11,27, 8,23)( 3,17,35,14,29,10,25, 6,21)( 4,18,36,13,30, 9,26, 5,22)$ |
| $ 18, 18 $ | $2$ | $18$ | $( 1,20,34,15,32,11,28, 8,24, 2,19,33,16,31,12,27, 7,23)( 3,18,35,13,29, 9,25, 5,21, 4,17,36,14,30,10,26, 6,22)$ |
Group invariants
| Order: | $36=2^{2} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [36, 1] |
| Character table: |
2 2 2 2 2 1 1 1 1 1 1 1 1
3 2 2 . . 2 2 2 2 2 2 2 2
1a 2a 4a 4b 9a 18a 18b 9b 6a 3a 9c 18c
2P 1a 1a 2a 2a 9b 9b 9c 9c 3a 3a 9a 9a
3P 1a 2a 4b 4a 3a 6a 6a 3a 2a 1a 3a 6a
5P 1a 2a 4a 4b 9c 18c 18a 9a 6a 3a 9b 18b
7P 1a 2a 4b 4a 9b 18b 18c 9c 6a 3a 9a 18a
11P 1a 2a 4b 4a 9b 18b 18c 9c 6a 3a 9a 18a
13P 1a 2a 4a 4b 9c 18c 18a 9a 6a 3a 9b 18b
17P 1a 2a 4a 4b 9a 18a 18b 9b 6a 3a 9c 18c
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 1 -1 A -A 1 -1 -1 1 -1 1 1 -1
X.4 1 -1 -A A 1 -1 -1 1 -1 1 1 -1
X.5 2 2 . . -1 -1 -1 -1 2 2 -1 -1
X.6 2 -2 . . -1 1 1 -1 -2 2 -1 1
X.7 2 2 . . B B D D -1 -1 C C
X.8 2 2 . . C C B B -1 -1 D D
X.9 2 2 . . D D C C -1 -1 B B
X.10 2 -2 . . B -B -D D 1 -1 C -C
X.11 2 -2 . . C -C -B B 1 -1 D -D
X.12 2 -2 . . D -D -C C 1 -1 B -B
A = -E(4)
= -Sqrt(-1) = -i
B = E(9)^2+E(9)^7
C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7
D = E(9)^4+E(9)^5
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