Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $32$ | |
| Group : | $C_2.S_3^2$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,36,8,4,33,5,2,35,7,3,34,6)(9,29,15,21,18,28,10,30,16,22,17,27)(11,31,13,24,19,25,12,32,14,23,20,26), (1,14)(2,13)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12)(17,35)(18,36)(19,33)(20,34)(21,30)(22,29)(23,31)(24,32) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 6: $S_3$ x 2 8: $C_4\times C_2$ 12: $D_{6}$ x 2 36: $S_3^2$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 2
Degree 4: $C_4$
Degree 6: $D_{6}$ x 2, $S_3^2$
Degree 9: $S_3^2$
Degree 12: $S_3 \times C_4$ x 2, 12T39
Degree 18: $S_3^2$
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, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,36)( 6,35)( 7,33)( 8,34)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,26) (16,25)(17,24)(18,23)(19,21)(20,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5,35)( 6,36)( 7,34)( 8,33)( 9,32)(10,31)(11,29)(12,30)(13,27) (14,28)(15,25)(16,26)(17,23)(18,24)(19,22)(20,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,22,10,21)(11,23,12,24)(13,26,14,25)(15,27,16,28) (17,30,18,29)(19,31,20,32)(33,36,34,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,36, 8,35)( 9,20,10,19)(11,18,12,17)(13,15,14,16) (21,31,22,32)(23,29,24,30)(25,28,26,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,21,10,22)(11,24,12,23)(13,25,14,26)(15,28,16,27) (17,29,18,30)(19,32,20,31)(33,35,34,36)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 2, 3)( 5,33, 6,34)( 7,35, 8,36)( 9,19,10,20)(11,17,12,18)(13,16,14,15) (21,32,22,31)(23,30,24,29)(25,27,26,28)$ |
| $ 12, 12, 12 $ | $6$ | $12$ | $( 1, 5,34, 4, 7,36, 2, 6,33, 3, 8,35)( 9,28,17,21,16,29,10,27,18,22,15,30) (11,25,20,24,14,31,12,26,19,23,13,32)$ |
| $ 12, 12, 12 $ | $6$ | $12$ | $( 1, 6,34, 3, 7,35, 2, 5,33, 4, 8,36)( 9,27,17,22,16,30,10,28,18,21,15,29) (11,26,20,23,14,32,12,25,19,24,13,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,33)( 2, 8,34)( 3, 5,36)( 4, 6,35)( 9,16,18)(10,15,17)(11,14,19) (12,13,20)(21,27,30)(22,28,29)(23,25,31)(24,26,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,33, 2, 7,34)( 3, 6,36, 4, 5,35)( 9,15,18,10,16,17)(11,13,19,12,14,20) (21,28,30,22,27,29)(23,26,31,24,25,32)$ |
| $ 12, 12, 12 $ | $6$ | $12$ | $( 1, 9,28,35,14,23, 2,10,27,36,13,24)( 3,12,26,33,16,22, 4,11,25,34,15,21) ( 5,20,32, 7,18,29, 6,19,31, 8,17,30)$ |
| $ 12, 12, 12 $ | $6$ | $12$ | $( 1,10,28,36,14,24, 2, 9,27,35,13,23)( 3,11,26,34,16,21, 4,12,25,33,15,22) ( 5,19,32, 8,18,30, 6,20,31, 7,17,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,11,30)( 2,12,29)( 3, 9,31)( 4,10,32)( 5,16,23)( 6,15,24)( 7,14,21) ( 8,13,22)(17,26,35)(18,25,36)(19,27,33)(20,28,34)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,12,30, 2,11,29)( 3,10,31, 4, 9,32)( 5,15,23, 6,16,24)( 7,13,21, 8,14,22) (17,25,35,18,26,36)(19,28,33,20,27,34)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,13,27, 2,14,28)( 3,15,25, 4,16,26)( 5,17,31, 6,18,32)( 7,20,30, 8,19,29) ( 9,24,36,10,23,35)(11,22,33,12,21,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,14,27)( 2,13,28)( 3,16,25)( 4,15,26)( 5,18,31)( 6,17,32)( 7,19,30) ( 8,20,29)( 9,23,36)(10,24,35)(11,21,33)(12,22,34)$ |
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [72, 21] |
| Character table: |
2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 1 1 2 2
3 2 . 2 . 1 1 1 1 1 1 2 2 1 1 2 2 2 2
1a 2a 2b 2c 4a 4b 4c 4d 12a 12b 3a 6a 12c 12d 3b 6b 6c 3c
2P 1a 1a 1a 1a 2b 2b 2b 2b 6a 6a 3a 3a 6c 6c 3b 3b 3c 3c
3P 1a 2a 2b 2c 4c 4d 4a 4b 4c 4a 1a 2b 4d 4b 1a 2b 2b 1a
5P 1a 2a 2b 2c 4a 4b 4c 4d 12a 12b 3a 6a 12c 12d 3b 6b 6c 3c
7P 1a 2a 2b 2c 4c 4d 4a 4b 12b 12a 3a 6a 12d 12c 3b 6b 6c 3c
11P 1a 2a 2b 2c 4c 4d 4a 4b 12b 12a 3a 6a 12d 12c 3b 6b 6c 3c
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 -1 -1 1 A -A -A A A -A 1 -1 -A A 1 -1 -1 1
X.6 1 -1 -1 1 -A A A -A -A A 1 -1 A -A 1 -1 -1 1
X.7 1 1 -1 -1 A A -A -A A -A 1 -1 A -A 1 -1 -1 1
X.8 1 1 -1 -1 -A -A A A -A A 1 -1 -A A 1 -1 -1 1
X.9 2 . 2 . . -2 . -2 . . 2 2 1 1 -1 -1 -1 -1
X.10 2 . 2 . . 2 . 2 . . 2 2 -1 -1 -1 -1 -1 -1
X.11 2 . 2 . -2 . -2 . 1 1 -1 -1 . . -1 -1 2 2
X.12 2 . 2 . 2 . 2 . -1 -1 -1 -1 . . -1 -1 2 2
X.13 2 . -2 . . B . -B . . 2 -2 -A A -1 1 1 -1
X.14 2 . -2 . . -B . B . . 2 -2 A -A -1 1 1 -1
X.15 2 . -2 . B . -B . -A A -1 1 . . -1 1 -2 2
X.16 2 . -2 . -B . B . A -A -1 1 . . -1 1 -2 2
X.17 4 . 4 . . . . . . . -2 -2 . . 1 1 -2 -2
X.18 4 . -4 . . . . . . . -2 2 . . 1 -1 2 -2
A = -E(4)
= -Sqrt(-1) = -i
B = -2*E(4)
= -2*Sqrt(-1) = -2i
|