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Magma
magma: G := TransitiveGroup(36, 38);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3:D_{12}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,30,16,5,26,19)(2,29,15,6,25,20)(3,31,14,7,28,17)(4,32,13,8,27,18)(9,36,24,10,35,23)(11,33,21)(12,34,22), (1,7,33,3,5,36,2,8,34,4,6,35)(9,26,17,21,14,30,10,25,18,22,13,29)(11,28,19,23,15,32,12,27,20,24,16,31) | 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 $24$: $D_{12}$, $(C_6\times C_2):C_2$ $36$: $S_3^2$ 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: $D_{12}$, $(C_6\times C_2):C_2$
Degree 18: $S_3^2$
Low degree siblings
12T38 x 2, 24T74, 36T33Siblings 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 $ | $6$ | $2$ | $( 3, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,17)(10,18)(11,20)(12,19)(13,14)(21,29) (22,30)(23,32)(24,31)(27,28)$ |
$ 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 $ | $18$ | $2$ | $( 1, 3)( 2, 4)( 5,35)( 6,36)( 7,33)( 8,34)( 9,30)(10,29)(11,31)(12,32)(13,25) (14,26)(15,27)(16,28)(17,21)(18,22)(19,24)(20,23)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,21,10,22)(11,23,12,24)(13,26,14,25)(15,27,16,28) (17,30,18,29)(19,32,20,31)(33,36,34,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5,34)( 2, 6,33)( 3, 8,35)( 4, 7,36)( 9,14,18)(10,13,17)(11,15,20) (12,16,19)(21,25,29)(22,26,30)(23,27,31)(24,28,32)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 6,34, 2, 5,33)( 3, 7,35, 4, 8,36)( 9,13,18,10,14,17)(11,16,20,12,15,19) (21,26,29,22,25,30)(23,28,31,24,27,32)$ |
$ 12, 12, 12 $ | $6$ | $12$ | $( 1, 7,33, 3, 5,36, 2, 8,34, 4, 6,35)( 9,26,17,21,14,30,10,25,18,22,13,29) (11,28,19,23,15,32,12,27,20,24,16,31)$ |
$ 12, 12, 12 $ | $6$ | $12$ | $( 1, 8,33, 4, 5,35, 2, 7,34, 3, 6,36)( 9,25,17,22,14,29,10,26,18,21,13,30) (11,27,19,24,15,31,12,28,20,23,16,32)$ |
$ 6, 6, 6, 6, 6, 3, 3 $ | $6$ | $6$ | $( 1,11,26,33,16,21)( 2,12,25,34,15,22)( 3, 9,28,35,14,24)( 4,10,27,36,13,23) ( 5,20,30, 6,19,29)( 7,17,31)( 8,18,32)$ |
$ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,11,30, 2,12,29)( 3,10,32, 4, 9,31)( 5,15,22, 6,16,21)( 7,14,23, 8,13,24) (17,28,36,18,27,35)(19,25,34,20,26,33)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,12,30)( 2,11,29)( 3, 9,32)( 4,10,31)( 5,16,22)( 6,15,21)( 7,13,23) ( 8,14,24)(17,27,36)(18,28,35)(19,26,34)(20,25,33)$ |
$ 6, 6, 6, 6, 6, 3, 3 $ | $6$ | $6$ | $( 1,12,26,34,16,22)( 2,11,25,33,15,21)( 3,10,28,36,14,23)( 4, 9,27,35,13,24) ( 5,19,30)( 6,20,29)( 7,18,31, 8,17,32)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,15,26, 2,16,25)( 3,13,28, 4,14,27)( 5,20,30, 6,19,29)( 7,18,31, 8,17,32) ( 9,23,35,10,24,36)(11,22,33,12,21,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,26)( 2,15,25)( 3,14,28)( 4,13,27)( 5,19,30)( 6,20,29)( 7,17,31) ( 8,18,32)( 9,24,35)(10,23,36)(11,21,33)(12,22,34)$ |
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: | 72.23 | 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 3a 6a 12a 12b 6b 6c 3b 6d 6e 3c 2P 1a 1a 1a 1a 2b 3a 3a 6a 6a 3c 3b 3b 3c 3c 3c 3P 1a 2a 2b 2c 4a 1a 2b 4a 4a 2a 2b 1a 2a 2b 1a 5P 1a 2a 2b 2c 4a 3a 6a 12b 12a 6d 6c 3b 6b 6e 3c 7P 1a 2a 2b 2c 4a 3a 6a 12b 12a 6b 6c 3b 6d 6e 3c 11P 1a 2a 2b 2c 4a 3a 6a 12a 12b 6d 6c 3b 6b 6e 3c 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 . . -1 1 A -A . 1 -1 . -2 2 X.11 2 . -2 . . -1 1 -A A . 1 -1 . -2 2 X.12 2 . -2 . . 2 -2 . . B 1 -1 -B 1 -1 X.13 2 . -2 . . 2 -2 . . -B 1 -1 B 1 -1 X.14 4 . 4 . . -2 -2 . . . 1 1 . -2 -2 X.15 4 . -4 . . -2 2 . . . -1 1 . 2 -2 A = -E(12)^7+E(12)^11 = Sqrt(3) = r3 B = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 |
magma: CharacterTable(G);