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Magma
magma: G := TransitiveGroup(36, 4);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_6^2$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $36$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,14,25,4,16,27)(2,13,26,3,15,28)(5,18,30,8,20,31)(6,17,29,7,19,32)(9,22,34,12,24,35)(10,21,33,11,23,36), (1,20,23,2,19,24)(3,17,22,4,18,21)(5,10,26,6,9,25)(7,12,27,8,11,28)(13,32,35,14,31,36)(15,29,34,16,30,33) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ x 4 $4$: $C_2^2$ $6$: $C_6$ x 12 $9$: $C_3^2$ $12$: $C_6\times C_2$ x 4 $18$: $C_6 \times C_3$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $C_3$ x 4
Degree 4: $C_2^2$
Degree 6: $C_6$ x 12
Degree 9: $C_3^2$
Degree 12: $C_6\times C_2$ x 4
Degree 18: $C_6 \times C_3$ x 3
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{36}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{18}$ | $1$ | $2$ | $18$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| 2B | $2^{18}$ | $1$ | $2$ | $18$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)$ |
| 2C | $2^{18}$ | $1$ | $2$ | $18$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,35)(34,36)$ |
| 3A1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1, 6,33)( 2, 5,34)( 3, 8,35)( 4, 7,36)( 9,15,20)(10,16,19)(11,14,17)(12,13,18)(21,27,32)(22,28,31)(23,25,29)(24,26,30)$ |
| 3A-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,10,29)( 2, 9,30)( 3,12,31)( 4,11,32)( 5,15,24)( 6,16,23)( 7,14,21)( 8,13,22)(17,27,36)(18,28,35)(19,25,33)(20,26,34)$ |
| 3B1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,16,25)( 2,15,26)( 3,13,28)( 4,14,27)( 5,20,30)( 6,19,29)( 7,17,32)( 8,18,31)( 9,24,34)(10,23,33)(11,21,36)(12,22,35)$ |
| 3B-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,29,10)( 2,30, 9)( 3,31,12)( 4,32,11)( 5,24,15)( 6,23,16)( 7,21,14)( 8,22,13)(17,36,27)(18,35,28)(19,33,25)(20,34,26)$ |
| 3C1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,25,16)( 2,26,15)( 3,28,13)( 4,27,14)( 5,30,20)( 6,29,19)( 7,32,17)( 8,31,18)( 9,34,24)(10,33,23)(11,36,21)(12,35,22)$ |
| 3C-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,23,19)( 2,24,20)( 3,22,18)( 4,21,17)( 5,26, 9)( 6,25,10)( 7,27,11)( 8,28,12)(13,35,31)(14,36,32)(15,34,30)(16,33,29)$ |
| 3D1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,19,23)( 2,20,24)( 3,18,22)( 4,17,21)( 5, 9,26)( 6,10,25)( 7,11,27)( 8,12,28)(13,31,35)(14,32,36)(15,30,34)(16,29,33)$ |
| 3D-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,33, 6)( 2,34, 5)( 3,35, 8)( 4,36, 7)( 9,20,15)(10,19,16)(11,17,14)(12,18,13)(21,32,27)(22,31,28)(23,29,25)(24,30,26)$ |
| 6A1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,28,16, 3,25,13)( 2,27,15, 4,26,14)( 5,32,20, 7,30,17)( 6,31,19, 8,29,18)( 9,36,24,11,34,21)(10,35,23,12,33,22)$ |
| 6A-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,15,25, 2,16,26)( 3,14,28, 4,13,27)( 5,19,30, 6,20,29)( 7,18,32, 8,17,31)( 9,23,34,10,24,33)(11,22,36,12,21,35)$ |
| 6B1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,24,19, 2,23,20)( 3,21,18, 4,22,17)( 5,25, 9, 6,26,10)( 7,28,11, 8,27,12)(13,36,31,14,35,32)(15,33,30,16,34,29)$ |
| 6B-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,14,25, 4,16,27)( 2,13,26, 3,15,28)( 5,18,30, 8,20,31)( 6,17,29, 7,19,32)( 9,22,34,12,24,35)(10,21,33,11,23,36)$ |
| 6C1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,35, 6, 3,33, 8)( 2,36, 5, 4,34, 7)( 9,17,15,11,20,14)(10,18,16,12,19,13)(21,30,27,24,32,26)(22,29,28,23,31,25)$ |
| 6C-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1, 9,29, 2,10,30)( 3,11,31, 4,12,32)( 5,16,24, 6,15,23)( 7,13,21, 8,14,22)(17,28,36,18,27,35)(19,26,33,20,25,34)$ |
| 6D1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,17,23, 4,19,21)( 2,18,24, 3,20,22)( 5,12,26, 8, 9,28)( 6,11,25, 7,10,27)(13,30,35,15,31,34)(14,29,36,16,32,33)$ |
| 6D-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,30,10, 2,29, 9)( 3,32,12, 4,31,11)( 5,23,15, 6,24,16)( 7,22,14, 8,21,13)(17,35,27,18,36,28)(19,34,25,20,33,26)$ |
| 6E1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,20,23, 2,19,24)( 3,17,22, 4,18,21)( 5,10,26, 6, 9,25)( 7,12,27, 8,11,28)(13,32,35,14,31,36)(15,29,34,16,30,33)$ |
| 6E-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,11,29, 4,10,32)( 2,12,30, 3, 9,31)( 5,13,24, 8,15,22)( 6,14,23, 7,16,21)(17,25,36,19,27,33)(18,26,35,20,28,34)$ |
| 6F1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,21,19, 4,23,17)( 2,22,20, 3,24,18)( 5,28, 9, 8,26,12)( 6,27,10, 7,25,11)(13,34,31,15,35,30)(14,33,32,16,36,29)$ |
| 6F-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,18,23, 3,19,22)( 2,17,24, 4,20,21)( 5,11,26, 7, 9,27)( 6,12,25, 8,10,28)(13,29,35,16,31,33)(14,30,36,15,32,34)$ |
| 6G1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,34, 6, 2,33, 5)( 3,36, 8, 4,35, 7)( 9,19,15,10,20,16)(11,18,14,12,17,13)(21,31,27,22,32,28)(23,30,25,24,29,26)$ |
| 6G-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,22,19, 3,23,18)( 2,21,20, 4,24,17)( 5,27, 9, 7,26,11)( 6,28,10, 8,25,12)(13,33,31,16,35,29)(14,34,32,15,36,30)$ |
| 6H1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,27,16, 4,25,14)( 2,28,15, 3,26,13)( 5,31,20, 8,30,18)( 6,32,19, 7,29,17)( 9,35,24,12,34,22)(10,36,23,11,33,21)$ |
| 6H-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,36, 6, 4,33, 7)( 2,35, 5, 3,34, 8)( 9,18,15,12,20,13)(10,17,16,11,19,14)(21,29,27,23,32,25)(22,30,28,24,31,26)$ |
| 6I1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,12,29, 3,10,31)( 2,11,30, 4, 9,32)( 5,14,24, 7,15,21)( 6,13,23, 8,16,22)(17,26,36,20,27,34)(18,25,35,19,28,33)$ |
| 6I-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,32,10, 4,29,11)( 2,31, 9, 3,30,12)( 5,22,15, 8,24,13)( 6,21,16, 7,23,14)(17,33,27,19,36,25)(18,34,28,20,35,26)$ |
| 6J1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1, 7,33, 4, 6,36)( 2, 8,34, 3, 5,35)( 9,13,20,12,15,18)(10,14,19,11,16,17)(21,25,32,23,27,29)(22,26,31,24,28,30)$ |
| 6J-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1, 5,33, 2, 6,34)( 3, 7,35, 4, 8,36)( 9,16,20,10,15,19)(11,13,17,12,14,18)(21,28,32,22,27,31)(23,26,29,24,25,30)$ |
| 6K1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,31,10, 3,29,12)( 2,32, 9, 4,30,11)( 5,21,15, 7,24,14)( 6,22,16, 8,23,13)(17,34,27,20,36,26)(18,33,28,19,35,25)$ |
| 6K-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1, 8,33, 3, 6,35)( 2, 7,34, 4, 5,36)( 9,14,20,11,15,17)(10,13,19,12,16,18)(21,26,32,24,27,30)(22,25,31,23,28,29)$ |
| 6L1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,26,16, 2,25,15)( 3,27,13, 4,28,14)( 5,29,20, 6,30,19)( 7,31,17, 8,32,18)( 9,33,24,10,34,23)(11,35,21,12,36,22)$ |
| 6L-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,13,25, 3,16,28)( 2,14,26, 4,15,27)( 5,17,30, 7,20,32)( 6,18,29, 8,19,31)( 9,21,34,11,24,36)(10,22,33,12,23,35)$ |
Malle's constant $a(G)$: $1/18$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $36=2^{2} \cdot 3^{2}$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 36.14 | magma: IdentifyGroup(G);
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| Character table: | 36 x 36 character table |
magma: CharacterTable(G);