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Magma
magma: G := TransitiveGroup(36, 17);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_4\times C_3^2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $18$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,27,15,4,26,13)(2,28,16,3,25,14)(5,32,20,8,29,17)(6,31,19,7,30,18)(9,35,23,12,33,21)(10,36,24,11,34,22), (1,9,29,2,10,30)(3,12,31)(4,11,32)(5,16,24,6,15,23)(7,14,21)(8,13,22)(17,27,36)(18,28,35)(19,26,33,20,25,34) | 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 $8$: $D_{4}$ $9$: $C_3^2$ $12$: $C_6\times C_2$ x 4 $18$: $C_6 \times C_3$ x 3 $24$: $D_4 \times C_3$ x 4 $36$: 36T4 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$ x 4
Degree 4: $D_{4}$
Degree 6: $C_6$ x 4
Degree 9: $C_3^2$
Degree 12: $D_4 \times C_3$ x 4
Degree 18: $C_6 \times C_3$
Low degree siblings
36T17Siblings are shown 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 | $2^{18}$ | $2$ | $2$ | $18$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)(33,35)(34,36)$ |
| 2C | $2^{9},1^{18}$ | $2$ | $2$ | $9$ | $( 3, 4)( 7, 8)(11,12)(13,14)(17,18)(21,22)(27,28)(31,32)(35,36)$ |
| 3A1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,26,15)( 2,25,16)( 3,28,14)( 4,27,13)( 5,29,20)( 6,30,19)( 7,31,18)( 8,32,17)( 9,33,23)(10,34,24)(11,36,22)(12,35,21)$ |
| 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 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,15,26)( 2,16,25)( 3,14,28)( 4,13,27)( 5,20,29)( 6,19,30)( 7,18,31)( 8,17,32)( 9,23,33)(10,24,34)(11,22,36)(12,21,35)$ |
| 3B-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1, 5,34)( 2, 6,33)( 3, 7,35)( 4, 8,36)( 9,16,19)(10,15,20)(11,13,17)(12,14,18)(21,28,31)(22,27,32)(23,25,30)(24,26,29)$ |
| 3C1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,24,20)( 2,23,19)( 3,21,18)( 4,22,17)( 5,26,10)( 6,25, 9)( 7,28,12)( 8,27,11)(13,36,32)(14,35,31)(15,34,29)(16,33,30)$ |
| 3C-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,20,24)( 2,19,23)( 3,18,21)( 4,17,22)( 5,10,26)( 6, 9,25)( 7,12,28)( 8,11,27)(13,32,36)(14,31,35)(15,29,34)(16,30,33)$ |
| 3D1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,29,10)( 2,30, 9)( 3,31,12)( 4,32,11)( 5,24,15)( 6,23,16)( 7,21,14)( 8,22,13)(17,36,27)(18,35,28)(19,33,25)(20,34,26)$ |
| 3D-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,34, 5)( 2,33, 6)( 3,35, 7)( 4,36, 8)( 9,19,16)(10,20,15)(11,17,13)(12,18,14)(21,31,28)(22,32,27)(23,30,25)(24,29,26)$ |
| 4A | $4^{9}$ | $2$ | $4$ | $27$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,35,34,36)$ |
| 6A1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,19,24, 2,20,23)( 3,17,21, 4,18,22)( 5, 9,26, 6,10,25)( 7,11,28, 8,12,27)(13,31,36,14,32,35)(15,30,34,16,29,33)$ |
| 6A-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,23,20, 2,24,19)( 3,22,18, 4,21,17)( 5,25,10, 6,26, 9)( 7,27,12, 8,28,11)(13,35,32,14,36,31)(15,33,29,16,34,30)$ |
| 6B1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,25,15, 2,26,16)( 3,27,14, 4,28,13)( 5,30,20, 6,29,19)( 7,32,18, 8,31,17)( 9,34,23,10,33,24)(11,35,22,12,36,21)$ |
| 6B-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1, 9,29, 2,10,30)( 3,11,31, 4,12,32)( 5,16,24, 6,15,23)( 7,13,21, 8,14,22)(17,28,36,18,27,35)(19,26,33,20,25,34)$ |
| 6C1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,33, 5, 2,34, 6)( 3,36, 7, 4,35, 8)( 9,20,16,10,19,15)(11,18,13,12,17,14)(21,32,28,22,31,27)(23,29,25,24,30,26)$ |
| 6C-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1, 6,34, 2, 5,33)( 3, 8,35, 4, 7,36)( 9,15,19,10,16,20)(11,14,17,12,13,18)(21,27,31,22,28,32)(23,26,30,24,25,29)$ |
| 6D1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,16,26, 2,15,25)( 3,13,28, 4,14,27)( 5,19,29, 6,20,30)( 7,17,31, 8,18,32)( 9,24,33,10,23,34)(11,21,36,12,22,35)$ |
| 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 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,15,26)( 2,16,25)( 3,13,28, 4,14,27)( 5,20,29)( 6,19,30)( 7,17,31, 8,18,32)( 9,23,33)(10,24,34)(11,21,36,12,22,35)$ |
| 6E-1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,26,15)( 2,25,16)( 3,27,14, 4,28,13)( 5,29,20)( 6,30,19)( 7,32,18, 8,31,17)( 9,33,23)(10,34,24)(11,35,22,12,36,21)$ |
| 6F1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,22,20, 4,24,17)( 2,21,19, 3,23,18)( 5,27,10, 8,26,11)( 6,28, 9, 7,25,12)(13,34,32,15,36,29)(14,33,31,16,35,30)$ |
| 6F-1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,29,10)( 2,30, 9)( 3,32,12, 4,31,11)( 5,24,15)( 6,23,16)( 7,22,14, 8,21,13)(17,35,27,18,36,28)(19,33,25)(20,34,26)$ |
| 6G1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,10,29)( 2, 9,30)( 3,11,31, 4,12,32)( 5,15,24)( 6,16,23)( 7,13,21, 8,14,22)(17,28,36,18,27,35)(19,25,33)(20,26,34)$ |
| 6G-1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,24,20)( 2,23,19)( 3,22,18, 4,21,17)( 5,26,10)( 6,25, 9)( 7,27,12, 8,28,11)(13,35,32,14,36,31)(15,34,29)(16,33,30)$ |
| 6H1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,27,15, 4,26,13)( 2,28,16, 3,25,14)( 5,32,20, 8,29,17)( 6,31,19, 7,30,18)( 9,35,23,12,33,21)(10,36,24,11,34,22)$ |
| 6H-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1, 8,34, 4, 5,36)( 2, 7,33, 3, 6,35)( 9,14,19,12,16,18)(10,13,20,11,15,17)(21,25,31,23,28,30)(22,26,32,24,27,29)$ |
| 6I1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,17,24, 4,20,22)( 2,18,23, 3,19,21)( 5,11,26, 8,10,27)( 6,12,25, 7, 9,28)(13,29,36,15,32,34)(14,30,35,16,31,33)$ |
| 6I-1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1, 5,34)( 2, 6,33)( 3, 8,35, 4, 7,36)( 9,16,19)(10,15,20)(11,14,17,12,13,18)(21,27,31,22,28,32)(23,25,30)(24,26,29)$ |
| 6J1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,20,24)( 2,19,23)( 3,17,21, 4,18,22)( 5,10,26)( 6, 9,25)( 7,11,28, 8,12,27)(13,31,36,14,32,35)(15,29,34)(16,30,33)$ |
| 6J-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,32,10, 4,29,11)( 2,31, 9, 3,30,12)( 5,22,15, 8,24,13)( 6,21,16, 7,23,14)(17,34,27,20,36,26)(18,33,28,19,35,25)$ |
| 6K1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,11,29, 4,10,32)( 2,12,30, 3, 9,31)( 5,13,24, 8,15,22)( 6,14,23, 7,16,21)(17,26,36,20,27,34)(18,25,35,19,28,33)$ |
| 6K-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,36, 5, 4,34, 8)( 2,35, 6, 3,33, 7)( 9,18,16,12,19,14)(10,17,15,11,20,13)(21,30,28,23,31,25)(22,29,27,24,32,26)$ |
| 6L1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,34, 5)( 2,33, 6)( 3,36, 7, 4,35, 8)( 9,19,16)(10,20,15)(11,18,13,12,17,14)(21,32,28,22,31,27)(23,30,25)(24,29,26)$ |
| 6L-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,13,26, 4,15,27)( 2,14,25, 3,16,28)( 5,17,29, 8,20,32)( 6,18,30, 7,19,31)( 9,21,33,12,23,35)(10,22,34,11,24,36)$ |
| 12A1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,27,16, 3,26,13, 2,28,15, 4,25,14)( 5,32,19, 7,29,17, 6,31,20, 8,30,18)( 9,35,24,11,33,21,10,36,23,12,34,22)$ |
| 12A-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,13,25, 3,15,27, 2,14,26, 4,16,28)( 5,17,30, 7,20,32, 6,18,29, 8,19,31)( 9,21,34,11,23,35,10,22,33,12,24,36)$ |
| 12B1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,32, 9, 3,29,11, 2,31,10, 4,30,12)( 5,22,16, 7,24,13, 6,21,15, 8,23,14)(17,33,28,20,36,25,18,34,27,19,35,26)$ |
| 12B-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,36, 6, 3,34, 8, 2,35, 5, 4,33, 7)( 9,18,15,11,19,14,10,17,16,12,20,13)(21,29,27,23,31,26,22,30,28,24,32,25)$ |
| 12C1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,22,19, 3,24,17, 2,21,20, 4,23,18)( 5,27, 9, 7,26,11, 6,28,10, 8,25,12)(13,33,31,15,36,30,14,34,32,16,35,29)$ |
| 12C-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,11,30, 3,10,32, 2,12,29, 4, 9,31)( 5,13,23, 7,15,22, 6,14,24, 8,16,21)(17,25,35,20,27,33,18,26,36,19,28,34)$ |
| 12D1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1, 8,33, 3, 5,36, 2, 7,34, 4, 6,35)( 9,14,20,11,16,18,10,13,19,12,15,17)(21,26,32,23,28,29,22,25,31,24,27,30)$ |
| 12D-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,17,23, 3,20,22, 2,18,24, 4,19,21)( 5,11,25, 7,10,27, 6,12,26, 8, 9,28)(13,30,35,15,32,33,14,29,36,16,31,34)$ |
Malle's constant $a(G)$: $1/9$
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: | $2$ | ||
| Label: | 72.37 | magma: IdentifyGroup(G);
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| Character table: | 45 x 45 character table |
magma: CharacterTable(G);