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Magma
magma: G := TransitiveGroup(36, 23);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3:S_4$ | ||
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: | (1,30,10)(2,29,9)(3,31,12)(4,32,11)(5,23,14)(6,24,13)(7,21,15)(8,22,16)(17,33,28)(18,34,27)(19,35,25)(20,36,26), (1,21,18)(2,22,17)(3,24,19)(4,23,20)(5,26,12)(6,25,11)(7,28,9)(8,27,10)(13,36,32)(14,35,31)(15,33,30)(16,34,29), (1,3)(2,4)(5,34)(6,33)(7,35)(8,36)(9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ x 4 $18$: $C_3^2:C_2$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 4
Degree 4: None
Degree 6: $S_3$ x 4, $S_4$, $S_4$
Degree 9: $C_3^2:C_2$
Degree 12: $S_4$
Degree 18: $C_3^2 : C_2$, 18T37, 18T40
Low degree siblings
12T44 x 3, 18T37, 18T40, 24T79 x 3, 36T56Siblings 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 $ | $3$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(17,18)(21,22)(23,24)(25,26)(29,30)(33,34) (35,36)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,32)(10,31)(11,29)(12,30)(13,28) (14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ | $18$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,36)( 8,35)( 9,31,10,32)(11,30)(12,29) (13,27,14,28)(15,25,16,26)(17,23)(18,24)(19,21,20,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 7,34)( 2, 8,33)( 3, 5,36)( 4, 6,35)( 9,16,18)(10,15,17)(11,14,20) (12,13,19)(21,28,29)(22,27,30)(23,25,31)(24,26,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 9,30)( 2,10,29)( 3,11,32)( 4,12,31)( 5,13,23)( 6,14,24)( 7,15,22) ( 8,16,21)(17,28,34)(18,27,33)(19,26,35)(20,25,36)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3 $ | $6$ | $6$ | $( 1,15,28, 2,16,27)( 3,13,26)( 4,14,25)( 5,20,32, 6,19,31)( 7,17,29, 8,18,30) ( 9,21,34)(10,22,33)(11,24,35,12,23,36)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,28)( 2,15,27)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,18,29) ( 8,17,30)( 9,21,34)(10,22,33)(11,23,35)(12,24,36)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,17,21)( 2,18,22)( 3,19,23)( 4,20,24)( 5,11,26)( 6,12,25)( 7,10,27) ( 8, 9,28)(13,32,35)(14,31,36)(15,29,33)(16,30,34)$ |
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: | 72.43 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 2 2 . . 2 2 . 3 2 1 . . 2 2 1 2 2 1a 2a 2b 4a 3a 3b 6a 3c 3d 2P 1a 1a 1a 2a 3a 3b 3c 3c 3d 3P 1a 2a 2b 4a 1a 1a 2a 1a 1a 5P 1a 2a 2b 4a 3a 3b 6a 3c 3d X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 X.3 2 2 . . -1 -1 2 2 -1 X.4 2 2 . . 2 -1 -1 -1 -1 X.5 2 2 . . -1 -1 -1 -1 2 X.6 2 2 . . -1 2 -1 -1 -1 X.7 3 -1 -1 1 . . -1 3 . X.8 3 -1 1 -1 . . -1 3 . X.9 6 -2 . . . . 1 -3 . |
magma: CharacterTable(G);