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Magma
magma: G := TransitiveGroup(36, 27);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_{12}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,9,26,33,16,23,2,10,25,34,15,24)(3,11,28,36,14,22,4,12,27,35,13,21)(5,17,32,7,20,29,6,18,31,8,19,30), (1,6,36,3,8,34,2,5,35,4,7,33)(9,15,20,11,13,18,10,16,19,12,14,17)(21,27,29,23,26,31,22,28,30,24,25,32) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $S_3 \times C_4$, 24T2 $36$: $C_6\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 9: $S_3\times C_3$
Degree 12: $C_{12}$, $S_3 \times C_4$
Degree 18: $S_3 \times C_6$
Low degree siblings
24T65, 36T27Siblings 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 | $2^{18}$ | $3$ | $2$ | $18$ | $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32)$ |
| 2C | $2^{12},1^{12}$ | $3$ | $2$ | $12$ | $( 5,33)( 6,34)( 7,36)( 8,35)( 9,19)(10,20)(11,17)(12,18)(21,30)(22,29)(23,32)(24,31)$ |
| 3A1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,16,25)( 2,15,26)( 3,14,27)( 4,13,28)( 5,20,31)( 6,19,32)( 7,18,30)( 8,17,29)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$ |
| 3A-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,25,16)( 2,26,15)( 3,27,14)( 4,28,13)( 5,31,20)( 6,32,19)( 7,30,18)( 8,29,17)( 9,34,23)(10,33,24)(11,35,22)(12,36,21)$ |
| 3B | $3^{12}$ | $2$ | $3$ | $24$ | $( 1, 8,35)( 2, 7,36)( 3, 5,33)( 4, 6,34)( 9,13,19)(10,14,20)(11,16,17)(12,15,18)(21,26,30)(22,25,29)(23,28,32)(24,27,31)$ |
| 3C1 | $3^{12}$ | $2$ | $3$ | $24$ | $( 1,17,22)( 2,18,21)( 3,20,24)( 4,19,23)( 5,10,27)( 6, 9,28)( 7,12,26)( 8,11,25)(13,32,34)(14,31,33)(15,30,36)(16,29,35)$ |
| 3C-1 | $3^{12}$ | $2$ | $3$ | $24$ | $( 1,29,11)( 2,30,12)( 3,31,10)( 4,32, 9)( 5,24,14)( 6,23,13)( 7,21,15)( 8,22,16)(17,35,25)(18,36,26)(19,34,28)(20,33,27)$ |
| 4A1 | $4^{9}$ | $1$ | $4$ | $27$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,36,34,35)$ |
| 4A-1 | $4^{9}$ | $1$ | $4$ | $27$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,35,34,36)$ |
| 4B1 | $4^{9}$ | $3$ | $4$ | $27$ | $( 1,34, 2,33)( 3,35, 4,36)( 5, 8, 6, 7)( 9,15,10,16)(11,13,12,14)(17,19,18,20)(21,27,22,28)(23,26,24,25)(29,32,30,31)$ |
| 4B-1 | $4^{9}$ | $3$ | $4$ | $27$ | $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,11,10,12)(13,17,14,18)(15,19,16,20)(21,23,22,24)(25,31,26,32)(27,30,28,29)(33,36,34,35)$ |
| 6A1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,26,16, 2,25,15)( 3,28,14, 4,27,13)( 5,32,20, 6,31,19)( 7,29,18, 8,30,17)( 9,33,23,10,34,24)(11,36,22,12,35,21)$ |
| 6A-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,15,25, 2,16,26)( 3,13,27, 4,14,28)( 5,19,31, 6,20,32)( 7,17,30, 8,18,29)( 9,24,34,10,23,33)(11,21,35,12,22,36)$ |
| 6B | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,12,29, 2,11,30)( 3, 9,31, 4,10,32)( 5,13,24, 6,14,23)( 7,16,21, 8,15,22)(17,26,35,18,25,36)(19,27,34,20,28,33)$ |
| 6C1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,36, 8, 2,35, 7)( 3,34, 5, 4,33, 6)( 9,20,13,10,19,14)(11,18,16,12,17,15)(21,29,26,22,30,25)(23,31,28,24,32,27)$ |
| 6C-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,21,17, 2,22,18)( 3,23,20, 4,24,19)( 5,28,10, 6,27, 9)( 7,25,12, 8,26,11)(13,33,32,14,34,31)(15,35,30,16,36,29)$ |
| 6D1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,12,25,36,16,21)( 2,11,26,35,15,22)( 3, 9,27,34,14,23)( 4,10,28,33,13,24)( 5,19,31, 6,20,32)( 7,17,30, 8,18,29)$ |
| 6D-1 | $6^{4},3^{4}$ | $3$ | $6$ | $28$ | $( 1,16,25)( 2,15,26)( 3,14,27)( 4,13,28)( 5,10,31,33,20,24)( 6, 9,32,34,19,23)( 7,12,30,36,18,21)( 8,11,29,35,17,22)$ |
| 6E1 | $6^{4},3^{4}$ | $3$ | $6$ | $28$ | $( 1,25,16)( 2,26,15)( 3,27,14)( 4,28,13)( 5,24,20,33,31,10)( 6,23,19,34,32, 9)( 7,21,18,36,30,12)( 8,22,17,35,29,11)$ |
| 6E-1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,21,16,36,25,12)( 2,22,15,35,26,11)( 3,23,14,34,27, 9)( 4,24,13,33,28,10)( 5,32,20, 6,31,19)( 7,29,18, 8,30,17)$ |
| 12A1 | $12^{3}$ | $1$ | $12$ | $33$ | $( 1,13,26, 3,16,28, 2,14,25, 4,15,27)( 5,17,32, 7,20,29, 6,18,31, 8,19,30)( 9,21,33,11,23,36,10,22,34,12,24,35)$ |
| 12A-1 | $12^{3}$ | $1$ | $12$ | $33$ | $( 1,27,15, 4,25,14, 2,28,16, 3,26,13)( 5,30,19, 8,31,18, 6,29,20, 7,32,17)( 9,35,24,12,34,22,10,36,23,11,33,21)$ |
| 12A5 | $12^{3}$ | $1$ | $12$ | $33$ | $( 1,28,15, 3,25,13, 2,27,16, 4,26,14)( 5,29,19, 7,31,17, 6,30,20, 8,32,18)( 9,36,24,11,34,21,10,35,23,12,33,22)$ |
| 12A-5 | $12^{3}$ | $1$ | $12$ | $33$ | $( 1,14,26, 4,16,27, 2,13,25, 3,15,28)( 5,18,32, 8,20,30, 6,17,31, 7,19,29)( 9,22,33,12,23,35,10,21,34,11,24,36)$ |
| 12B1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1, 6,36, 3, 8,34, 2, 5,35, 4, 7,33)( 9,15,20,11,13,18,10,16,19,12,14,17)(21,27,29,23,26,31,22,28,30,24,25,32)$ |
| 12B-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,31,12, 4,29,10, 2,32,11, 3,30, 9)( 5,21,13, 8,24,15, 6,22,14, 7,23,16)(17,33,26,19,35,27,18,34,25,20,36,28)$ |
| 12C1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,19,21, 3,17,23, 2,20,22, 4,18,24)( 5,11,28, 7,10,25, 6,12,27, 8, 9,26)(13,30,33,16,32,36,14,29,34,15,31,35)$ |
| 12C-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,20,21, 4,17,24, 2,19,22, 3,18,23)( 5,12,28, 8,10,26, 6,11,27, 7, 9,25)(13,29,33,15,32,35,14,30,34,16,31,36)$ |
| 12C5 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,32,12, 3,29, 9, 2,31,11, 4,30,10)( 5,22,13, 7,24,16, 6,21,14, 8,23,15)(17,34,26,20,35,28,18,33,25,19,36,27)$ |
| 12C-5 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1, 5,36, 4, 8,33, 2, 6,35, 3, 7,34)( 9,16,20,12,13,17,10,15,19,11,14,18)(21,28,29,24,26,32,22,27,30,23,25,31)$ |
| 12D1 | $12^{3}$ | $3$ | $12$ | $33$ | $( 1,23,15,33,25, 9, 2,24,16,34,26,10)( 3,22,13,36,27,11, 4,21,14,35,28,12)( 5,29,19, 7,31,17, 6,30,20, 8,32,18)$ |
| 12D-1 | $12^{3}$ | $3$ | $12$ | $33$ | $( 1,31,15, 6,25,20, 2,32,16, 5,26,19)( 3,30,13, 8,27,18, 4,29,14, 7,28,17)( 9,35,24,12,34,22,10,36,23,11,33,21)$ |
| 12D5 | $12^{3}$ | $3$ | $12$ | $33$ | $( 1,20,26, 6,16,31, 2,19,25, 5,15,32)( 3,18,28, 8,14,30, 4,17,27, 7,13,29)( 9,22,33,12,23,35,10,21,34,11,24,36)$ |
| 12D-5 | $12^{3}$ | $3$ | $12$ | $33$ | $( 1, 9,26,33,16,23, 2,10,25,34,15,24)( 3,11,28,36,14,22, 4,12,27,35,13,21)( 5,17,32, 7,20,29, 6,18,31, 8,19,30)$ |
Malle's constant $a(G)$: $1/12$
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: | 72.27 | magma: IdentifyGroup(G);
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| Character table: | 36 x 36 character table |
magma: CharacterTable(G);