Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $73$ | |
| Group : | $C_3\times C_3:S_3.C_2$ | |
| CHM label : | $1/2[3^{3}:2]4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,8,7,2)(3,6,9,12)(4,11,10,5), (2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) | |
| $|\Aut(F/K)|$: | $3$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $C_6$ 12: $C_{12}$ 36: $C_3^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: None
Low degree siblings
12T73, 18T44 x 2, 27T33, 36T81 x 2, 36T95 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 3, 7,11)( 4,12, 8)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 3,11, 7)( 4, 8,12)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 6,10)( 4,12, 8)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $4$ | $3$ | $( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $4$ | $3$ | $( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
| $ 12 $ | $9$ | $12$ | $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$ |
| $ 12 $ | $9$ | $12$ | $( 1, 2, 3, 8, 9, 6,11,12, 5,10, 7, 4)$ |
| $ 4, 4, 4 $ | $9$ | $4$ | $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$ |
| $ 6, 6 $ | $9$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$ |
| $ 6, 6 $ | $9$ | $6$ | $( 1, 3, 9,11, 5, 7)( 2, 4, 6, 8,10,12)$ |
| $ 12 $ | $9$ | $12$ | $( 1, 4, 7,10, 5,12,11, 6, 9, 8, 3, 2)$ |
| $ 12 $ | $9$ | $12$ | $( 1, 4, 3, 6, 9, 8,11,10, 5,12, 7, 2)$ |
| $ 4, 4, 4 $ | $9$ | $4$ | $( 1, 4,11, 2)( 3,10, 5,12)( 6, 9, 8, 7)$ |
| $ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$ |
| $ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ |
| $ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$ |
Group invariants
| Order: | $108=2^{2} \cdot 3^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [108, 36] |
| Character table: |
2 2 . . . . . 2 2 2 2 2 2 2 2 2 . 2 2
3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 3 3 3
1a 3a 3b 3c 3d 3e 12a 12b 4a 2a 6a 6b 12c 12d 4b 3f 3g 3h
2P 1a 3b 3a 3c 3e 3d 6a 6b 2a 1a 3g 3h 6a 6b 2a 3f 3h 3g
3P 1a 1a 1a 1a 1a 1a 4b 4b 4b 2a 2a 2a 4a 4a 4a 1a 1a 1a
5P 1a 3b 3a 3c 3e 3d 12b 12a 4a 2a 6b 6a 12d 12c 4b 3f 3h 3g
7P 1a 3a 3b 3c 3d 3e 12c 12d 4b 2a 6a 6b 12a 12b 4a 3f 3g 3h
11P 1a 3b 3a 3c 3e 3d 12d 12c 4b 2a 6b 6a 12b 12a 4a 3f 3h 3g
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 C C C -1 -1 -1 -C -C -C 1 1 1
X.4 1 1 1 1 1 1 -C -C -C -1 -1 -1 C C C 1 1 1
X.5 1 A /A 1 A /A -/A -A -1 1 A /A -/A -A -1 1 /A A
X.6 1 /A A 1 /A A -A -/A -1 1 /A A -A -/A -1 1 A /A
X.7 1 A /A 1 A /A /A A 1 1 A /A /A A 1 1 /A A
X.8 1 /A A 1 /A A A /A 1 1 /A A A /A 1 1 A /A
X.9 1 A /A 1 A /A D -/D C -1 -A -/A -D /D -C 1 /A A
X.10 1 A /A 1 A /A -D /D -C -1 -A -/A D -/D C 1 /A A
X.11 1 /A A 1 /A A -/D D C -1 -/A -A /D -D -C 1 A /A
X.12 1 /A A 1 /A A /D -D -C -1 -/A -A -/D D C 1 A /A
X.13 4 -2 -2 1 1 1 . . . . . . . . . -2 4 4
X.14 4 1 1 -2 -2 -2 . . . . . . . . . 1 4 4
X.15 4 B /B 1 /A A . . . . . . . . . -2 E /E
X.16 4 /B B 1 A /A . . . . . . . . . -2 /E E
X.17 4 /A A -2 B /B . . . . . . . . . 1 E /E
X.18 4 A /A -2 /B B . . . . . . . . . 1 /E E
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = -2*E(3)
= 1-Sqrt(-3) = 1-i3
C = -E(4)
= -Sqrt(-1) = -i
D = -E(12)^7
E = 4*E(3)^2
= -2-2*Sqrt(-3) = -2-2i3
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