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Magma
magma: G := TransitiveGroup(36, 25);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2:D_9$ | ||
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: | (1,22,2,21)(3,23)(4,24)(5,20,6,19)(7,18,8,17)(9,16,10,15)(11,14)(12,13)(25,34)(26,33)(27,35,28,36)(29,31,30,32), (1,33,32,27,24,20,16,11,7)(2,34,31,28,23,19,15,12,8)(3,35,30,25,22,17,13,10,5)(4,36,29,26,21,18,14,9,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $18$: $D_{9}$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: None
Degree 9: $D_{9}$
Degree 12: $S_4$
Low degree siblings
18T38, 18T39, 36T57Siblings 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 $ | $3$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(19,20)(21,22)(23,24)(25,26)(31,32)(33,34) (35,36)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,31)(10,32)(11,30)(12,29)(13,27) (14,28)(15,26)(16,25)(17,24)(18,23)(19,21)(20,22)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ | $18$ | $4$ | $( 1, 3, 2, 4)( 5,33, 6,34)( 7,36)( 8,35)( 9,32)(10,31)(11,29,12,30) (13,28,14,27)(15,26,16,25)(17,24,18,23)(19,22)(20,21)$ |
$ 9, 9, 9, 9 $ | $8$ | $9$ | $( 1, 7,11,16,20,24,27,32,33)( 2, 8,12,15,19,23,28,31,34)( 3, 5,10,13,17,22,25, 30,35)( 4, 6, 9,14,18,21,26,29,36)$ |
$ 9, 9, 9, 9 $ | $8$ | $9$ | $( 1,11,19,27,33, 8,16,24,31)( 2,12,20,28,34, 7,15,23,32)( 3, 9,17,25,36, 5,13, 21,30)( 4,10,18,26,35, 6,14,22,29)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3 $ | $6$ | $6$ | $( 1,15,27, 2,16,28)( 3,13,25)( 4,14,26)( 5,18,30, 6,17,29)( 7,19,32, 8,20,31) ( 9,22,36,10,21,35)(11,24,33)(12,23,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,27)( 2,15,28)( 3,13,25)( 4,14,26)( 5,17,30)( 6,18,29)( 7,20,32) ( 8,19,31)( 9,21,36)(10,22,35)(11,24,33)(12,23,34)$ |
$ 9, 9, 9, 9 $ | $8$ | $9$ | $( 1,19,33,16,31,11,27, 8,24)( 2,20,34,15,32,12,28, 7,23)( 3,17,36,13,30, 9,25, 5,21)( 4,18,35,14,29,10,26, 6,22)$ |
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: | 72.15 | 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 9a 9b 6a 3a 9c 2P 1a 1a 1a 2a 9b 9c 3a 3a 9a 3P 1a 2a 2b 4a 3a 3a 2a 1a 3a 5P 1a 2a 2b 4a 9c 9a 6a 3a 9b 7P 1a 2a 2b 4a 9b 9c 6a 3a 9a 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 . . A C -1 -1 B X.5 2 2 . . B A -1 -1 C X.6 2 2 . . C B -1 -1 A X.7 3 -1 -1 1 . . -1 3 . X.8 3 -1 1 -1 . . -1 3 . X.9 6 -2 . . . . 1 -3 . A = E(9)^4+E(9)^5 B = E(9)^2+E(9)^7 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7 |
magma: CharacterTable(G);