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Magma
magma: G := TransitiveGroup(36, 32);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6.D_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,36,8,4,33,5,2,35,7,3,34,6)(9,29,15,21,18,28,10,30,16,22,17,27)(11,31,13,24,19,25,12,32,14,23,20,26), (1,14)(2,13)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12)(17,35)(18,36)(19,33)(20,34)(21,30)(22,29)(23,31)(24,32) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ x 2 $8$: $C_4\times C_2$ $12$: $D_{6}$ x 2 $24$: $S_3 \times C_4$ 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: $C_4$
Degree 6: $D_{6}$ x 2, $S_3^2$
Degree 9: $S_3^2$
Degree 12: $S_3 \times C_4$ x 2, 12T39
Degree 18: $S_3^2$
Low degree siblings
12T39 x 2, 24T75, 36T32Siblings 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, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,36)( 6,35)( 7,33)( 8,34)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,26) (16,25)(17,24)(18,23)(19,21)(20,22)$ |
$ 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 $ | $9$ | $2$ | $( 1, 2)( 3, 4)( 5,35)( 6,36)( 7,34)( 8,33)( 9,32)(10,31)(11,29)(12,30)(13,27) (14,28)(15,25)(16,26)(17,23)(18,24)(19,22)(20,21)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,22,10,21)(11,23,12,24)(13,26,14,25)(15,27,16,28) (17,30,18,29)(19,31,20,32)(33,36,34,35)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 3, 2, 4)( 5,34, 6,33)( 7,36, 8,35)( 9,20,10,19)(11,18,12,17)(13,15,14,16) (21,31,22,32)(23,29,24,30)(25,28,26,27)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,21,10,22)(11,24,12,23)(13,25,14,26)(15,28,16,27) (17,29,18,30)(19,32,20,31)(33,35,34,36)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 2, 3)( 5,33, 6,34)( 7,35, 8,36)( 9,19,10,20)(11,17,12,18)(13,16,14,15) (21,32,22,31)(23,30,24,29)(25,27,26,28)$ |
$ 12, 12, 12 $ | $6$ | $12$ | $( 1, 5,34, 4, 7,36, 2, 6,33, 3, 8,35)( 9,28,17,21,16,29,10,27,18,22,15,30) (11,25,20,24,14,31,12,26,19,23,13,32)$ |
$ 12, 12, 12 $ | $6$ | $12$ | $( 1, 6,34, 3, 7,35, 2, 5,33, 4, 8,36)( 9,27,17,22,16,30,10,28,18,21,15,29) (11,26,20,23,14,32,12,25,19,24,13,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,33)( 2, 8,34)( 3, 5,36)( 4, 6,35)( 9,16,18)(10,15,17)(11,14,19) (12,13,20)(21,27,30)(22,28,29)(23,25,31)(24,26,32)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1, 8,33, 2, 7,34)( 3, 6,36, 4, 5,35)( 9,15,18,10,16,17)(11,13,19,12,14,20) (21,28,30,22,27,29)(23,26,31,24,25,32)$ |
$ 12, 12, 12 $ | $6$ | $12$ | $( 1, 9,28,35,14,23, 2,10,27,36,13,24)( 3,12,26,33,16,22, 4,11,25,34,15,21) ( 5,20,32, 7,18,29, 6,19,31, 8,17,30)$ |
$ 12, 12, 12 $ | $6$ | $12$ | $( 1,10,28,36,14,24, 2, 9,27,35,13,23)( 3,11,26,34,16,21, 4,12,25,33,15,22) ( 5,19,32, 8,18,30, 6,20,31, 7,17,29)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,11,30)( 2,12,29)( 3, 9,31)( 4,10,32)( 5,16,23)( 6,15,24)( 7,14,21) ( 8,13,22)(17,26,35)(18,25,36)(19,27,33)(20,28,34)$ |
$ 6, 6, 6, 6, 6, 6 $ | $4$ | $6$ | $( 1,12,30, 2,11,29)( 3,10,31, 4, 9,32)( 5,15,23, 6,16,24)( 7,13,21, 8,14,22) (17,25,35,18,26,36)(19,28,33,20,27,34)$ |
$ 6, 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,13,27, 2,14,28)( 3,15,25, 4,16,26)( 5,17,31, 6,18,32)( 7,20,30, 8,19,29) ( 9,24,36,10,23,35)(11,22,33,12,21,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,14,27)( 2,13,28)( 3,16,25)( 4,15,26)( 5,18,31)( 6,17,32)( 7,19,30) ( 8,20,29)( 9,23,36)(10,24,35)(11,21,33)(12,22,34)$ |
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: | 72.21 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 1 1 2 2 3 2 . 2 . 1 1 1 1 1 1 2 2 1 1 2 2 2 2 1a 2a 2b 2c 4a 4b 4c 4d 12a 12b 3a 6a 12c 12d 3b 6b 6c 3c 2P 1a 1a 1a 1a 2b 2b 2b 2b 6a 6a 3a 3a 6c 6c 3b 3b 3c 3c 3P 1a 2a 2b 2c 4c 4d 4a 4b 4c 4a 1a 2b 4d 4b 1a 2b 2b 1a 5P 1a 2a 2b 2c 4a 4b 4c 4d 12a 12b 3a 6a 12c 12d 3b 6b 6c 3c 7P 1a 2a 2b 2c 4c 4d 4a 4b 12b 12a 3a 6a 12d 12c 3b 6b 6c 3c 11P 1a 2a 2b 2c 4c 4d 4a 4b 12b 12a 3a 6a 12d 12c 3b 6b 6c 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 A -A -A A A -A 1 -1 -A A 1 -1 -1 1 X.6 1 -1 -1 1 -A A A -A -A A 1 -1 A -A 1 -1 -1 1 X.7 1 1 -1 -1 A A -A -A A -A 1 -1 A -A 1 -1 -1 1 X.8 1 1 -1 -1 -A -A A A -A A 1 -1 -A A 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 . 1 1 -1 -1 . . -1 -1 2 2 X.12 2 . 2 . 2 . 2 . -1 -1 -1 -1 . . -1 -1 2 2 X.13 2 . -2 . . B . -B . . 2 -2 -A A -1 1 1 -1 X.14 2 . -2 . . -B . B . . 2 -2 A -A -1 1 1 -1 X.15 2 . -2 . B . -B . -A A -1 1 . . -1 1 -2 2 X.16 2 . -2 . -B . B . A -A -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 A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i |
magma: CharacterTable(G);