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Magma
magma: G := TransitiveGroup(27, 33);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:C_{12}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,6,21,18,2,4,19,16,3,5,20,17)(7,22,14,26,8,23,15,27,9,24,13,25)(10,11,12), (1,3,2)(4,16,25,13,6,18,27,15,5,17,26,14)(7,19,22,11,9,21,24,10,8,20,23,12) | magma: Generators(G);
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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 3: $C_3$
Degree 9: $C_3^2:C_4$
Low degree siblings
12T73 x 2, 18T44 x 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $9$ | $4$ | $( 4,13,27,17)( 5,14,25,18)( 6,15,26,16)( 7,11,24,20)( 8,12,22,21)( 9,10,23,19)$ |
$ 4, 4, 4, 4, 4, 4, 1, 1, 1 $ | $9$ | $4$ | $( 4,17,27,13)( 5,18,25,14)( 6,16,26,15)( 7,20,24,11)( 8,21,22,12)( 9,19,23,10)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 4,27)( 5,25)( 6,26)( 7,24)( 8,22)( 9,23)(10,19)(11,20)(12,21)(13,17)(14,18) (15,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 12, 12, 3 $ | $9$ | $12$ | $( 1, 2, 3)( 4,14,26,17, 5,15,27,18, 6,13,25,16)( 7,12,23,20, 8,10,24,21, 9,11, 22,19)$ |
$ 12, 12, 3 $ | $9$ | $12$ | $( 1, 2, 3)( 4,18,26,13, 5,16,27,14, 6,17,25,15)( 7,21,23,11, 8,19,24,12, 9,20, 22,10)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4,25, 6,27, 5,26)( 7,22, 9,24, 8,23)(10,20,12,19,11,21) (13,18,15,17,14,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
$ 12, 12, 3 $ | $9$ | $12$ | $( 1, 3, 2)( 4,15,25,17, 6,14,27,16, 5,13,26,18)( 7,10,22,20, 9,12,24,19, 8,11, 23,21)$ |
$ 12, 12, 3 $ | $9$ | $12$ | $( 1, 3, 2)( 4,16,25,13, 6,18,27,15, 5,17,26,14)( 7,19,22,11, 9,21,24,10, 8,20, 23,12)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4,26, 5,27, 6,25)( 7,23, 8,24, 9,22)(10,21,11,19,12,20) (13,16,14,17,15,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,10,13)( 8,11,14)( 9,12,15)(16,19,22) (17,20,23)(18,21,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24) (17,21,22)(18,19,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,12,14)( 8,10,15)( 9,11,13)(16,21,23) (17,19,24)(18,20,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 7,22)( 2, 8,23)( 3, 9,24)( 4,10,16)( 5,11,17)( 6,12,18)(13,19,26) (14,20,27)(15,21,25)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 8,24)( 2, 9,22)( 3, 7,23)( 4,11,18)( 5,12,16)( 6,10,17)(13,20,25) (14,21,26)(15,19,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,23)( 2, 7,24)( 3, 8,22)( 4,12,17)( 5,10,18)( 6,11,16)(13,21,27) (14,19,25)(15,20,26)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 108.36 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 . . . . . . 3 3 1 1 1 3 1 1 1 3 1 1 1 3 3 3 3 3 3 1a 4a 4b 2a 3a 12a 12b 6a 3b 12c 12d 6b 3c 3d 3e 3f 3g 3h 2P 1a 2a 2a 1a 3b 6b 6b 3b 3a 6a 6a 3a 3e 3d 3c 3g 3f 3h 3P 1a 4b 4a 2a 1a 4b 4a 2a 1a 4b 4a 2a 1a 1a 1a 1a 1a 1a 5P 1a 4a 4b 2a 3b 12c 12d 6b 3a 12a 12b 6a 3e 3d 3c 3g 3f 3h 7P 1a 4b 4a 2a 3a 12b 12a 6a 3b 12d 12c 6b 3c 3d 3e 3f 3g 3h 11P 1a 4b 4a 2a 3b 12d 12c 6b 3a 12b 12a 6a 3e 3d 3c 3g 3f 3h 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 B -B -B B /B -/B -/B /B /B 1 B B /B 1 X.4 1 -1 -1 1 /B -/B -/B /B B -B -B B B 1 /B /B B 1 X.5 1 1 1 1 B B B B /B /B /B /B /B 1 B B /B 1 X.6 1 1 1 1 /B /B /B /B B B B B B 1 /B /B B 1 X.7 1 A -A -1 1 A -A -1 1 A -A -1 1 1 1 1 1 1 X.8 1 -A A -1 1 -A A -1 1 -A A -1 1 1 1 1 1 1 X.9 1 A -A -1 B D -D -B /B -/D /D -/B /B 1 B B /B 1 X.10 1 A -A -1 /B -/D /D -/B B D -D -B B 1 /B /B B 1 X.11 1 -A A -1 B -D D -B /B /D -/D -/B /B 1 B B /B 1 X.12 1 -A A -1 /B /D -/D -/B B -D D -B B 1 /B /B B 1 X.13 4 . . . 4 . . . 4 . . . -2 -2 -2 1 1 1 X.14 4 . . . 4 . . . 4 . . . 1 1 1 -2 -2 -2 X.15 4 . . . C . . . /C . . . E -2 /E B /B 1 X.16 4 . . . /C . . . C . . . /E -2 E /B B 1 X.17 4 . . . C . . . /C . . . /B 1 B /E E -2 X.18 4 . . . /C . . . C . . . B 1 /B E /E -2 A = -E(4) = -Sqrt(-1) = -i B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 C = 4*E(3)^2 = -2-2*Sqrt(-3) = -2-2i3 D = -E(12)^11 E = -2*E(3) = 1-Sqrt(-3) = 1-i3 |
magma: CharacterTable(G);