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Magma
magma: G := TransitiveGroup(36, 40);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times D_6$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,35,6,4,33,8)(2,36,5,3,34,7)(9,29,13,22,18,26)(10,30,14,21,17,25)(11,31,16,24,20,27)(12,32,15,23,19,28), (1,34)(2,33)(3,35)(4,36)(5,6)(7,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,30)(31,32), (1,9)(2,10)(3,12)(4,11)(5,17)(6,18)(7,19)(8,20)(13,33)(14,34)(15,36)(16,35)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ x 2 $8$: $C_2^3$ $12$: $D_{6}$ x 6 $24$: $S_3 \times C_2^2$ x 2 $36$: $S_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $S_3$ x 2
Degree 4: $C_2^2$
Degree 9: $S_3^2$
Degree 12: $D_6$, $S_3 \times C_2^2$
Low degree siblings
12T37 x 2, 18T29 x 4, 24T73, 36T34 x 2, 36T40 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 5,34)( 6,33)( 7,36)( 8,35)( 9,18)(10,17)(11,20)(12,19)(21,30)(22,29)(23,32) (24,31)$ |
$ 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 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,17)(10,18)(11,19)(12,20)(13,14) (15,16)(21,29)(22,30)(23,31)(24,32)(25,26)(27,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27) (16,28)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5,35)( 6,36)( 7,33)( 8,34)( 9,30)(10,29)(11,32)(12,31)(13,25) (14,26)(15,27)(16,28)(17,22)(18,21)(19,24)(20,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,22)(10,21)(11,24)(12,23)(13,26)(14,25)(15,28) (16,27)(17,30)(18,29)(19,32)(20,31)(33,35)(34,36)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2, 3)( 5,36)( 6,35)( 7,34)( 8,33)( 9,29)(10,30)(11,31)(12,32)(13,26) (14,25)(15,28)(16,27)(17,21)(18,22)(19,23)(20,24)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 5,33, 2, 6,34)( 3, 8,36, 4, 7,35)( 9,14,18,10,13,17)(11,15,20,12,16,19) (21,26,30,22,25,29)(23,27,32,24,28,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6,33)( 2, 5,34)( 3, 7,36)( 4, 8,35)( 9,13,18)(10,14,17)(11,16,20) (12,15,19)(21,25,30)(22,26,29)(23,28,32)(24,27,31)$ |
$ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 7,33, 3, 6,36)( 2, 8,34, 4, 5,35)( 9,25,18,21,13,30)(10,26,17,22,14,29) (11,28,20,23,16,32)(12,27,19,24,15,31)$ |
$ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 8,33, 4, 6,35)( 2, 7,34, 3, 5,36)( 9,26,18,22,13,29)(10,25,17,21,14,30) (11,27,20,24,16,31)(12,28,19,23,15,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,11,29)( 2,12,30)( 3,10,32)( 4, 9,31)( 5,15,21)( 6,16,22)( 7,14,23) ( 8,13,24)(17,28,36)(18,27,35)(19,25,34)(20,26,33)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3 $ | $6$ | $6$ | $( 1,11,26,33,16,22)( 2,12,25,34,15,21)( 3,10,28,36,14,23)( 4, 9,27,35,13,24) ( 5,19,30)( 6,20,29)( 7,17,32)( 8,18,31)$ |
$ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,12,29, 2,11,30)( 3, 9,32, 4,10,31)( 5,16,21, 6,15,22)( 7,13,23, 8,14,24) (17,27,36,18,28,35)(19,26,34,20,25,33)$ |
$ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1,12,26,34,16,21)( 2,11,25,33,15,22)( 3, 9,28,35,14,24)( 4,10,27,36,13,23) ( 5,20,30, 6,19,29)( 7,18,32, 8,17,31)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,26, 2,16,25)( 3,13,28, 4,14,27)( 5,20,30, 6,19,29)( 7,18,32, 8,17,31) ( 9,23,35,10,24,36)(11,21,33,12,22,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,26)( 2,15,25)( 3,14,28)( 4,13,27)( 5,19,30)( 6,20,29)( 7,17,32) ( 8,18,31)( 9,24,35)(10,23,36)(11,22,33)(12,21,34)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.46 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 3 3 3 3 3 3 2 2 2 2 1 2 1 2 2 2 3 2 1 2 1 1 . 1 . 2 2 1 1 2 1 2 1 2 2 1a 2a 2b 2c 2d 2e 2f 2g 6a 3a 6b 6c 3b 6d 6e 6f 6g 3c 2P 1a 1a 1a 1a 1a 1a 1a 1a 3a 3a 3a 3a 3b 3c 3b 3c 3c 3c 3P 1a 2a 2b 2c 2d 2e 2f 2g 2b 1a 2d 2f 1a 2a 2b 2c 2b 1a 5P 1a 2a 2b 2c 2d 2e 2f 2g 6a 3a 6b 6c 3b 6d 6e 6f 6g 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 1 -1 1 -1 1 1 1 1 1 -1 1 -1 1 1 X.6 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 X.7 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 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 . . . . -2 2 . . -1 -1 1 1 1 -1 X.12 2 2 2 2 . . . . 2 2 . . -1 -1 -1 -1 -1 -1 X.13 2 . -2 . -2 . 2 . 1 -1 1 -1 -1 . 1 . -2 2 X.14 2 . -2 . 2 . -2 . 1 -1 -1 1 -1 . 1 . -2 2 X.15 2 . 2 . -2 . -2 . -1 -1 1 1 -1 . -1 . 2 2 X.16 2 . 2 . 2 . 2 . -1 -1 -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 |
magma: CharacterTable(G);