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Magma
magma: G := TransitiveGroup(36, 39);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_6:S_3$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,8,34,4,6,35)(2,7,33,3,5,36)(9,26,18,22,14,29)(10,25,17,21,13,30)(11,27,20,23,15,31)(12,28,19,24,16,32), (1,24,2,23)(3,21,4,22)(5,31,6,32)(7,30,8,29)(9,15,10,16)(11,13,12,14)(17,19,18,20)(25,35,26,36)(27,34,28,33) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 2 $8$: $D_{4}$ $12$: $D_{6}$ x 2 $24$: $(C_6\times C_2):C_2$ x 2 $36$: $S_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 2
Degree 4: $D_{4}$
Degree 9: $S_3^2$
Degree 12: $(C_6\times C_2):C_2$, $(C_6\times C_2):C_2$
Degree 18: $S_3^2$
Low degree siblings
24T61, 36T39Siblings 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, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 3, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,17)(10,18)(11,20)(12,19)(13,14)(21,30) (22,29)(23,32)(24,31)(27,28)$ |
$ 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 $ | $6$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,21)(10,22)(11,24)(12,23)(13,26)(14,25)(15,28) (16,27)(17,29)(18,30)(19,31)(20,32)(33,35)(34,36)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $18$ | $4$ | $( 1, 3, 2, 4)( 5,35, 6,36)( 7,33, 8,34)( 9,29,10,30)(11,32,12,31)(13,25,14,26) (15,28,16,27)(17,21,18,22)(19,23,20,24)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 5,34, 2, 6,33)( 3, 8,36, 4, 7,35)( 9,13,18,10,14,17)(11,16,20,12,15,19) (21,26,30,22,25,29)(23,28,31,24,27,32)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6,34)( 2, 5,33)( 3, 7,36)( 4, 8,35)( 9,14,18)(10,13,17)(11,15,20) (12,16,19)(21,25,30)(22,26,29)(23,27,31)(24,28,32)$ |
$ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 7,34, 3, 6,36)( 2, 8,33, 4, 5,35)( 9,25,18,21,14,30)(10,26,17,22,13,29) (11,28,20,24,15,32)(12,27,19,23,16,31)$ |
$ 6, 6, 6, 6, 6, 6 $ | $6$ | $6$ | $( 1, 8,34, 4, 6,35)( 2, 7,33, 3, 5,36)( 9,26,18,22,14,29)(10,25,17,21,13,30) (11,27,20,23,15,31)(12,28,19,24,16,32)$ |
$ 6, 6, 6, 6, 6, 3, 3 $ | $6$ | $6$ | $( 1,11,26,34,15,22)( 2,12,25,33,16,21)( 3, 9,28,35,13,23)( 4,10,27,36,14,24) ( 5,19,30)( 6,20,29)( 7,18,32, 8,17,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,11,29)( 2,12,30)( 3,10,32)( 4, 9,31)( 5,16,21)( 6,15,22)( 7,13,24) ( 8,14,23)(17,28,36)(18,27,35)(19,25,33)(20,26,34)$ |
$ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,12,29, 2,11,30)( 3, 9,32, 4,10,31)( 5,15,21, 6,16,22)( 7,14,24, 8,13,23) (17,27,36,18,28,35)(19,26,33,20,25,34)$ |
$ 6, 6, 6, 6, 6, 3, 3 $ | $6$ | $6$ | $( 1,12,26,33,15,21)( 2,11,25,34,16,22)( 3,10,28,36,13,24)( 4, 9,27,35,14,23) ( 5,20,30, 6,19,29)( 7,17,32)( 8,18,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,15,26)( 2,16,25)( 3,13,28)( 4,14,27)( 5,19,30)( 6,20,29)( 7,17,32) ( 8,18,31)( 9,23,35)(10,24,36)(11,22,34)(12,21,33)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,16,26, 2,15,25)( 3,14,28, 4,13,27)( 5,20,30, 6,19,29)( 7,18,32, 8,17,31) ( 9,24,35,10,23,36)(11,21,34,12,22,33)$ |
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: | 72.22 | magma: IdentifyGroup(G);
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Character table: |
2 3 2 3 2 2 2 2 2 2 2 1 1 2 2 2 3 2 1 2 1 . 2 2 1 1 1 2 2 1 2 2 1a 2a 2b 2c 4a 6a 3a 6b 6c 6d 3b 6e 6f 3c 6g 2P 1a 1a 1a 1a 2b 3a 3a 3a 3a 3c 3b 3b 3c 3c 3c 3P 1a 2a 2b 2c 4a 2b 1a 2c 2c 2a 1a 2b 2a 1a 2b 5P 1a 2a 2b 2c 4a 6a 3a 6c 6b 6f 3b 6e 6d 3c 6g X.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 X.3 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 X.5 2 -2 2 . . 2 2 . . 1 -1 -1 1 -1 -1 X.6 2 2 2 . . 2 2 . . -1 -1 -1 -1 -1 -1 X.7 2 . -2 . . -2 2 . . . 2 -2 . 2 -2 X.8 2 . 2 -2 . -1 -1 1 1 . -1 -1 . 2 2 X.9 2 . 2 2 . -1 -1 -1 -1 . -1 -1 . 2 2 X.10 2 . -2 . . -2 2 . . A -1 1 -A -1 1 X.11 2 . -2 . . -2 2 . . -A -1 1 A -1 1 X.12 2 . -2 . . 1 -1 A -A . -1 1 . 2 -2 X.13 2 . -2 . . 1 -1 -A A . -1 1 . 2 -2 X.14 4 . -4 . . 2 -2 . . . 1 -1 . -2 2 X.15 4 . 4 . . -2 -2 . . . 1 1 . -2 -2 A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 |
magma: CharacterTable(G);