Group action invariants
| Degree $n$ : | $36$ | |
| Transitive number $t$ : | $33$ | |
| Group : | $C_3^2:D_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,13)(8,14)(9,36)(10,35)(11,34)(12,33)(21,22)(25,31)(26,32)(27,29)(28,30), (1,23,7,25,33,31)(2,24,8,26,34,32)(3,21,5,27,36,30)(4,22,6,28,35,29)(9,19,16,11,18,14)(10,20,15,12,17,13) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ x 2 8: $D_{4}$ 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: $D_{4}$
Degree 6: $D_{6}$ x 2, $S_3^2$
Degree 9: $S_3^2$
Degree 12: $D_{12}$, $(C_6\times C_2):C_2$, 12T38 x 2
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, 2, 1, 1 $ | $18$ | $2$ | $( 3, 4)( 5,35)( 6,36)( 7,33)( 8,34)( 9,32)(10,31)(11,30)(12,29)(13,28)(14,27) (15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,28) (16,27)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,36, 8,35)( 9,20,10,19)(11,18,12,17)(13,15,14,16) (21,31,22,32)(23,29,24,30)(25,28,26,27)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,33, 3, 7,36)( 2, 6,34, 4, 8,35)( 9,27,18,21,16,30)(10,28,17,22,15,29) (11,25,19,23,14,31)(12,26,20,24,13,32)$ |
| $ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 6,33, 4, 7,35)( 2, 5,34, 3, 8,36)( 9,28,18,22,16,29)(10,27,17,21,15,30) (11,26,19,24,14,32)(12,25,20,23,13,31)$ |
| $ 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, 23] |
| Character table: |
2 3 2 3 2 2 2 2 2 2 2 2 1 1 2 2
3 2 . 2 1 1 1 1 2 2 1 1 2 2 2 2
1a 2a 2b 2c 4a 6a 6b 3a 6c 12a 12b 3b 6d 6e 3c
2P 1a 1a 1a 1a 2b 3a 3a 3a 3a 6e 6e 3b 3b 3c 3c
3P 1a 2a 2b 2c 4a 2c 2c 1a 2b 4a 4a 1a 2b 2b 1a
5P 1a 2a 2b 2c 4a 6b 6a 3a 6c 12b 12a 3b 6d 6e 3c
7P 1a 2a 2b 2c 4a 6a 6b 3a 6c 12b 12a 3b 6d 6e 3c
11P 1a 2a 2b 2c 4a 6b 6a 3a 6c 12a 12b 3b 6d 6e 3c
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 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
X.5 2 . 2 . -2 . . 2 2 1 1 -1 -1 -1 -1
X.6 2 . 2 . 2 . . 2 2 -1 -1 -1 -1 -1 -1
X.7 2 . -2 . . . . 2 -2 . . 2 -2 -2 2
X.8 2 . 2 -2 . 1 1 -1 -1 . . -1 -1 2 2
X.9 2 . 2 2 . -1 -1 -1 -1 . . -1 -1 2 2
X.10 2 . -2 . . . . 2 -2 B -B -1 1 1 -1
X.11 2 . -2 . . . . 2 -2 -B B -1 1 1 -1
X.12 2 . -2 . . A -A -1 1 . . -1 1 -2 2
X.13 2 . -2 . . -A A -1 1 . . -1 1 -2 2
X.14 4 . 4 . . . . -2 -2 . . 1 1 -2 -2
X.15 4 . -4 . . . . -2 2 . . 1 -1 2 -2
A = -E(3)+E(3)^2
= -Sqrt(-3) = -i3
B = -E(12)^7+E(12)^11
= Sqrt(3) = r3
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