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Magma
magma: G := TransitiveGroup(36, 15);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_4\times C_9$ | ||
| 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: | (1,15,27,4,13,26)(2,16,28,3,14,25)(5,17,32,8,20,29)(6,18,31,7,19,30)(9,21,33,11,23,35)(10,22,34,12,24,36), (1,30,22,16,6,33,28,17,11,4,32,24,13,7,36,25,19,9,2,29,21,15,5,34,27,18,12,3,31,23,14,8,35,26,20,10) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $8$: $D_{4}$ $9$: $C_9$ $12$: $C_6\times C_2$ $18$: $C_{18}$ x 3 $24$: $D_4 \times C_3$ $36$: 36T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 4: $D_{4}$
Degree 6: $C_6$
Degree 9: $C_9$
Degree 12: $D_4 \times C_3$
Degree 18: $C_{18}$
Low degree siblings
36T15Siblings 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 | $2^{9},1^{18}$ | $2$ | $2$ | $9$ | $( 3, 4)( 7, 8)( 9,10)(15,16)(17,18)(23,24)(25,26)(29,30)(33,34)$ |
| 2C | $2^{18}$ | $2$ | $2$ | $18$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)(33,35)(34,36)$ |
| 3A1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,13,27)( 2,14,28)( 3,16,25)( 4,15,26)( 5,20,32)( 6,19,31)( 7,18,30)( 8,17,29)( 9,23,33)(10,24,34)(11,21,35)(12,22,36)$ |
| 3A-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,27,13)( 2,28,14)( 3,25,16)( 4,26,15)( 5,32,20)( 6,31,19)( 7,30,18)( 8,29,17)( 9,33,23)(10,34,24)(11,35,21)(12,36,22)$ |
| 4A | $4^{9}$ | $2$ | $4$ | $27$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,36,34,35)$ |
| 6A1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,14,27, 2,13,28)( 3,15,25, 4,16,26)( 5,19,32, 6,20,31)( 7,17,30, 8,18,29)( 9,24,33,10,23,34)(11,22,35,12,21,36)$ |
| 6A-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,28,13, 2,27,14)( 3,26,16, 4,25,15)( 5,31,20, 6,32,19)( 7,29,18, 8,30,17)( 9,34,23,10,33,24)(11,36,21,12,35,22)$ |
| 6B1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,26,13, 4,27,15)( 2,25,14, 3,28,16)( 5,29,20, 8,32,17)( 6,30,19, 7,31,18)( 9,35,23,11,33,21)(10,36,24,12,34,22)$ |
| 6B-1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,27,13)( 2,28,14)( 3,26,16, 4,25,15)( 5,32,20)( 6,31,19)( 7,29,18, 8,30,17)( 9,34,23,10,33,24)(11,35,21)(12,36,22)$ |
| 6C1 | $6^{3},3^{6}$ | $2$ | $6$ | $27$ | $( 1,13,27)( 2,14,28)( 3,15,25, 4,16,26)( 5,20,32)( 6,19,31)( 7,17,30, 8,18,29)( 9,24,33,10,23,34)(11,21,35)(12,22,36)$ |
| 6C-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,15,27, 4,13,26)( 2,16,28, 3,14,25)( 5,17,32, 8,20,29)( 6,18,31, 7,19,30)( 9,21,33,11,23,35)(10,22,34,12,24,36)$ |
| 9A1 | $9^{4}$ | $1$ | $9$ | $32$ | $( 1,35,31,27,21,19,13,11, 6)( 2,36,32,28,22,20,14,12, 5)( 3,34,29,25,24,17,16,10, 8)( 4,33,30,26,23,18,15, 9, 7)$ |
| 9A-1 | $9^{4}$ | $1$ | $9$ | $32$ | $( 1,21, 6,27,11,31,13,35,19)( 2,22, 5,28,12,32,14,36,20)( 3,24, 8,25,10,29,16,34,17)( 4,23, 7,26, 9,30,15,33,18)$ |
| 9A2 | $9^{4}$ | $1$ | $9$ | $32$ | $( 1,19,35,13,31,11,27, 6,21)( 2,20,36,14,32,12,28, 5,22)( 3,17,34,16,29,10,25, 8,24)( 4,18,33,15,30, 9,26, 7,23)$ |
| 9A-2 | $9^{4}$ | $1$ | $9$ | $32$ | $( 1,11,19,27,35, 6,13,21,31)( 2,12,20,28,36, 5,14,22,32)( 3,10,17,25,34, 8,16,24,29)( 4, 9,18,26,33, 7,15,23,30)$ |
| 9A4 | $9^{4}$ | $1$ | $9$ | $32$ | $( 1,31,21,13, 6,35,27,19,11)( 2,32,22,14, 5,36,28,20,12)( 3,29,24,16, 8,34,25,17,10)( 4,30,23,15, 7,33,26,18, 9)$ |
| 9A-4 | $9^{4}$ | $1$ | $9$ | $32$ | $( 1, 6,11,13,19,21,27,31,35)( 2, 5,12,14,20,22,28,32,36)( 3, 8,10,16,17,24,25,29,34)( 4, 7, 9,15,18,23,26,30,33)$ |
| 12A1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,26,14, 3,27,15, 2,25,13, 4,28,16)( 5,29,19, 7,32,17, 6,30,20, 8,31,18)( 9,36,24,11,33,22,10,35,23,12,34,21)$ |
| 12A-1 | $12^{3}$ | $2$ | $12$ | $33$ | $( 1,15,28, 3,13,26, 2,16,27, 4,14,25)( 5,17,31, 7,20,29, 6,18,32, 8,19,30)( 9,22,34,11,23,36,10,21,33,12,24,35)$ |
| 18A1 | $18^{2}$ | $1$ | $18$ | $34$ | $( 1,12,19,28,35, 5,13,22,31, 2,11,20,27,36, 6,14,21,32)( 3, 9,17,26,34, 7,16,23,29, 4,10,18,25,33, 8,15,24,30)$ |
| 18A-1 | $18^{2}$ | $1$ | $18$ | $34$ | $( 1,22, 6,28,11,32,13,36,19, 2,21, 5,27,12,31,14,35,20)( 3,23, 8,26,10,30,16,33,17, 4,24, 7,25, 9,29,15,34,18)$ |
| 18A5 | $18^{2}$ | $1$ | $18$ | $34$ | $( 1,20,35,14,31,12,27, 5,21, 2,19,36,13,32,11,28, 6,22)( 3,18,34,15,29, 9,25, 7,24, 4,17,33,16,30,10,26, 8,23)$ |
| 18A-5 | $18^{2}$ | $1$ | $18$ | $34$ | $( 1,36,31,28,21,20,13,12, 6, 2,35,32,27,22,19,14,11, 5)( 3,33,29,26,24,18,16, 9, 8, 4,34,30,25,23,17,15,10, 7)$ |
| 18A7 | $18^{2}$ | $1$ | $18$ | $34$ | $( 1, 5,11,14,19,22,27,32,35, 2, 6,12,13,20,21,28,31,36)( 3, 7,10,15,17,23,25,30,34, 4, 8, 9,16,18,24,26,29,33)$ |
| 18A-7 | $18^{2}$ | $1$ | $18$ | $34$ | $( 1,32,21,14, 6,36,27,20,11, 2,31,22,13, 5,35,28,19,12)( 3,30,24,15, 8,33,25,18,10, 4,29,23,16, 7,34,26,17, 9)$ |
| 18B1 | $18,9^{2}$ | $2$ | $18$ | $33$ | $( 1, 6,11,13,19,21,27,31,35)( 2, 5,12,14,20,22,28,32,36)( 3, 7,10,15,17,23,25,30,34, 4, 8, 9,16,18,24,26,29,33)$ |
| 18B-1 | $18,9^{2}$ | $2$ | $18$ | $33$ | $( 1,31,21,13, 6,35,27,19,11)( 2,32,22,14, 5,36,28,20,12)( 3,30,24,15, 8,33,25,18,10, 4,29,23,16, 7,34,26,17, 9)$ |
| 18B5 | $18^{2}$ | $2$ | $18$ | $34$ | $( 1,23, 6,26,11,30,13,33,19, 4,21, 7,27, 9,31,15,35,18)( 2,24, 5,25,12,29,14,34,20, 3,22, 8,28,10,32,16,36,17)$ |
| 18B-5 | $18,9^{2}$ | $2$ | $18$ | $33$ | $( 1,19,35,13,31,11,27, 6,21)( 2,20,36,14,32,12,28, 5,22)( 3,18,34,15,29, 9,25, 7,24, 4,17,33,16,30,10,26, 8,23)$ |
| 18B7 | $18,9^{2}$ | $2$ | $18$ | $33$ | $( 1,11,19,27,35, 6,13,21,31)( 2,12,20,28,36, 5,14,22,32)( 3, 9,17,26,34, 7,16,23,29, 4,10,18,25,33, 8,15,24,30)$ |
| 18B-7 | $18^{2}$ | $2$ | $18$ | $34$ | $( 1, 9,19,26,35, 7,13,23,31, 4,11,18,27,33, 6,15,21,30)( 2,10,20,25,36, 8,14,24,32, 3,12,17,28,34, 5,16,22,29)$ |
| 18C1 | $18^{2}$ | $2$ | $18$ | $34$ | $( 1,30,21,15, 6,33,27,18,11, 4,31,23,13, 7,35,26,19, 9)( 2,29,22,16, 5,34,28,17,12, 3,32,24,14, 8,36,25,20,10)$ |
| 18C-1 | $18^{2}$ | $2$ | $18$ | $34$ | $( 1,18,35,15,31, 9,27, 7,21, 4,19,33,13,30,11,26, 6,23)( 2,17,36,16,32,10,28, 8,22, 3,20,34,14,29,12,25, 5,24)$ |
| 18C5 | $18^{2}$ | $2$ | $18$ | $34$ | $( 1, 7,11,15,19,23,27,30,35, 4, 6, 9,13,18,21,26,31,33)( 2, 8,12,16,20,24,28,29,36, 3, 5,10,14,17,22,25,32,34)$ |
| 18C-5 | $18^{2}$ | $2$ | $18$ | $34$ | $( 1,33,31,26,21,18,13, 9, 6, 4,35,30,27,23,19,15,11, 7)( 2,34,32,25,22,17,14,10, 5, 3,36,29,28,24,20,16,12, 8)$ |
| 18C7 | $18,9^{2}$ | $2$ | $18$ | $33$ | $( 1,35,31,27,21,19,13,11, 6)( 2,36,32,28,22,20,14,12, 5)( 3,33,29,26,24,18,16, 9, 8, 4,34,30,25,23,17,15,10, 7)$ |
| 18C-7 | $18,9^{2}$ | $2$ | $18$ | $33$ | $( 1,21, 6,27,11,31,13,35,19)( 2,22, 5,28,12,32,14,36,20)( 3,23, 8,26,10,30,16,33,17, 4,24, 7,25, 9,29,15,34,18)$ |
| 36A1 | $36$ | $2$ | $36$ | $35$ | $( 1, 7,12,16,19,23,28,29,35, 4, 5,10,13,18,22,25,31,33, 2, 8,11,15,20,24,27,30,36, 3, 6, 9,14,17,21,26,32,34)$ |
| 36A-1 | $36$ | $2$ | $36$ | $35$ | $( 1,30,22,16, 6,33,28,17,11, 4,32,24,13, 7,36,25,19, 9, 2,29,21,15, 5,34,27,18,12, 3,31,23,14, 8,35,26,20,10)$ |
| 36A5 | $36$ | $2$ | $36$ | $35$ | $( 1,18,36,16,31, 9,28, 8,21, 4,20,34,13,30,12,25, 6,23, 2,17,35,15,32,10,27, 7,22, 3,19,33,14,29,11,26, 5,24)$ |
| 36A-5 | $36$ | $2$ | $36$ | $35$ | $( 1, 9,20,25,35, 7,14,24,31, 4,12,17,27,33, 5,16,21,30, 2,10,19,26,36, 8,13,23,32, 3,11,18,28,34, 6,15,22,29)$ |
| 36A7 | $36$ | $2$ | $36$ | $35$ | $( 1,33,32,25,21,18,14,10, 6, 4,36,29,27,23,20,16,11, 7, 2,34,31,26,22,17,13, 9, 5, 3,35,30,28,24,19,15,12, 8)$ |
| 36A-7 | $36$ | $2$ | $36$ | $35$ | $( 1,23, 5,25,11,30,14,34,19, 4,22, 8,27, 9,32,16,35,18, 2,24, 6,26,12,29,13,33,20, 3,21, 7,28,10,31,15,36,17)$ |
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: | 72.10 | magma: IdentifyGroup(G);
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| Character table: | 45 x 45 character table |
magma: CharacterTable(G);