Show commands:
Magma
magma: G := TransitiveGroup(36, 29);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_6\times D_6$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,29,15,7,25,17)(2,30,16,8,26,18)(3,31,13,6,28,19)(4,32,14,5,27,20)(9,33,24)(10,34,23)(11,36,22)(12,35,21), (1,5)(2,6)(3,8)(4,7)(9,12)(10,11)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,32)(26,31)(27,29)(28,30)(33,35)(34,36), (1,16,25,2,15,26)(3,14,28,4,13,27)(5,10,32,34,20,23)(6,9,31,33,19,24)(7,11,29,36,17,22)(8,12,30,35,18,21) | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $3$: $C_3$ $4$: $C_2^2$ x 7 $6$: $S_3$, $C_6$ x 7 $8$: $C_2^3$ $12$: $D_{6}$ x 3, $C_6\times C_2$ x 7 $18$: $S_3\times C_3$ $24$: $S_3 \times C_2^2$, 24T3 $36$: $C_6\times S_3$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 6: $C_6$ x 3, $D_{6}$ x 3
Degree 9: $S_3\times C_3$
Degree 12: $C_6\times C_2$, $S_3 \times C_2^2$
Degree 18: $S_3 \times C_6$ x 3
Low degree siblings
24T68, 36T29 x 3Siblings 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 | $2^{18}$ | $1$ | $2$ | $18$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)$ |
| 2B | $2^{18}$ | $1$ | $2$ | $18$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$ |
| 2C | $2^{18}$ | $1$ | $2$ | $18$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$ |
| 2D | $2^{18}$ | $3$ | $2$ | $18$ | $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,19)(10,20)(11,17)(12,18)(13,14)(15,16)(21,30)(22,29)(23,32)(24,31)(25,26)(27,28)$ |
| 2E | $2^{18}$ | $3$ | $2$ | $18$ | $( 1,34)( 2,33)( 3,35)( 4,36)( 5, 8)( 6, 7)( 9,16)(10,15)(11,14)(12,13)(17,19)(18,20)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32)$ |
| 2F | $2^{12},1^{12}$ | $3$ | $2$ | $12$ | $( 5,33)( 6,34)( 7,35)( 8,36)( 9,20)(10,19)(11,18)(12,17)(21,29)(22,30)(23,31)(24,32)$ |
| 2G | $2^{18}$ | $3$ | $2$ | $18$ | $( 1,33)( 2,34)( 3,36)( 4,35)( 5, 7)( 6, 8)( 9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,27)(22,28)(23,26)(24,25)(29,32)(30,31)$ |
| 3A1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,15,25)( 2,16,26)( 3,13,28)( 4,14,27)( 5,20,32)( 6,19,31)( 7,17,29)( 8,18,30)( 9,24,33)(10,23,34)(11,22,36)(12,21,35)$ |
| 3A-1 | $3^{12}$ | $1$ | $3$ | $24$ | $( 1,25,15)( 2,26,16)( 3,28,13)( 4,27,14)( 5,32,20)( 6,31,19)( 7,29,17)( 8,30,18)( 9,33,24)(10,34,23)(11,36,22)(12,35,21)$ |
| 3B | $3^{12}$ | $2$ | $3$ | $24$ | $( 1,35, 7)( 2,36, 8)( 3,34, 6)( 4,33, 5)( 9,20,14)(10,19,13)(11,18,16)(12,17,15)(21,29,25)(22,30,26)(23,31,28)(24,32,27)$ |
| 3C1 | $3^{12}$ | $2$ | $3$ | $24$ | $( 1,21,17)( 2,22,18)( 3,23,19)( 4,24,20)( 5,27, 9)( 6,28,10)( 7,25,12)( 8,26,11)(13,34,31)(14,33,32)(15,35,29)(16,36,30)$ |
| 3C-1 | $3^{12}$ | $2$ | $3$ | $24$ | $( 1,12,29)( 2,11,30)( 3,10,31)( 4, 9,32)( 5,14,24)( 6,13,23)( 7,15,21)( 8,16,22)(17,25,35)(18,26,36)(19,28,34)(20,27,33)$ |
| 6A1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,14,25, 4,15,27)( 2,13,26, 3,16,28)( 5,17,32, 7,20,29)( 6,18,31, 8,19,30)( 9,21,33,12,24,35)(10,22,34,11,23,36)$ |
| 6A-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,13,25, 3,15,28)( 2,14,26, 4,16,27)( 5,18,32, 8,20,30)( 6,17,31, 7,19,29)( 9,22,33,11,24,36)(10,21,34,12,23,35)$ |
| 6B1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,28,15, 3,25,13)( 2,27,16, 4,26,14)( 5,30,20, 8,32,18)( 6,29,19, 7,31,17)( 9,36,24,11,33,22)(10,35,23,12,34,21)$ |
| 6B-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,26,15, 2,25,16)( 3,27,13, 4,28,14)( 5,31,20, 6,32,19)( 7,30,17, 8,29,18)( 9,34,24,10,33,23)(11,35,22,12,36,21)$ |
| 6C1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,27,15, 4,25,14)( 2,28,16, 3,26,13)( 5,29,20, 7,32,17)( 6,30,19, 8,31,18)( 9,35,24,12,33,21)(10,36,23,11,34,22)$ |
| 6C-1 | $6^{6}$ | $1$ | $6$ | $30$ | $( 1,16,25, 2,15,26)( 3,14,28, 4,13,27)( 5,19,32, 6,20,31)( 7,18,29, 8,17,30)( 9,23,33,10,24,34)(11,21,36,12,22,35)$ |
| 6D | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,36, 7, 2,35, 8)( 3,33, 6, 4,34, 5)( 9,19,14,10,20,13)(11,17,16,12,18,15)(21,30,25,22,29,26)(23,32,28,24,31,27)$ |
| 6E | $6^{6}$ | $2$ | $6$ | $30$ | $( 1, 5,35, 4, 7,33)( 2, 6,36, 3, 8,34)( 9,15,20,12,14,17)(10,16,19,11,13,18)(21,27,29,24,25,32)(22,28,30,23,26,31)$ |
| 6F | $6^{6}$ | $2$ | $6$ | $30$ | $( 1, 6,35, 3, 7,34)( 2, 5,36, 4, 8,33)( 9,16,20,11,14,18)(10,15,19,12,13,17)(21,28,29,23,25,31)(22,27,30,24,26,32)$ |
| 6G1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,22,17, 2,21,18)( 3,24,19, 4,23,20)( 5,28, 9, 6,27,10)( 7,26,12, 8,25,11)(13,33,31,14,34,32)(15,36,29,16,35,30)$ |
| 6G-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,20,21, 4,17,24)( 2,19,22, 3,18,23)( 5,12,27, 7, 9,25)( 6,11,28, 8,10,26)(13,30,34,16,31,36)(14,29,33,15,32,35)$ |
| 6H1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,11,29, 2,12,30)( 3, 9,31, 4,10,32)( 5,13,24, 6,14,23)( 7,16,21, 8,15,22)(17,26,35,18,25,36)(19,27,34,20,28,33)$ |
| 6H-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,32,12, 4,29, 9)( 2,31,11, 3,30,10)( 5,21,14, 7,24,15)( 6,22,13, 8,23,16)(17,33,25,20,35,27)(18,34,26,19,36,28)$ |
| 6I1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,19,21, 3,17,23)( 2,20,22, 4,18,24)( 5,11,27, 8, 9,26)( 6,12,28, 7,10,25)(13,29,34,15,31,35)(14,30,33,16,32,36)$ |
| 6I-1 | $6^{6}$ | $2$ | $6$ | $30$ | $( 1,31,12, 3,29,10)( 2,32,11, 4,30, 9)( 5,22,14, 8,24,16)( 6,21,13, 7,23,15)(17,34,25,19,35,28)(18,33,26,20,36,27)$ |
| 6J1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1, 9,25,33,15,24)( 2,10,26,34,16,23)( 3,11,28,36,13,22)( 4,12,27,35,14,21)( 5,17,32, 7,20,29)( 6,18,31, 8,19,30)$ |
| 6J-1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,23,15,34,25,10)( 2,24,16,33,26, 9)( 3,21,13,35,28,12)( 4,22,14,36,27,11)( 5,30,20, 8,32,18)( 6,29,19, 7,31,17)$ |
| 6K1 | $6^{4},3^{4}$ | $3$ | $6$ | $28$ | $( 1,25,15)( 2,26,16)( 3,28,13)( 4,27,14)( 5,24,20,33,32, 9)( 6,23,19,34,31,10)( 7,21,17,35,29,12)( 8,22,18,36,30,11)$ |
| 6K-1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,16,25, 2,15,26)( 3,14,28, 4,13,27)( 5,10,32,34,20,23)( 6, 9,31,33,19,24)( 7,11,29,36,17,22)( 8,12,30,35,18,21)$ |
| 6L1 | $6^{4},3^{4}$ | $3$ | $6$ | $28$ | $( 1,15,25)( 2,16,26)( 3,13,28)( 4,14,27)( 5, 9,32,33,20,24)( 6,10,31,34,19,23)( 7,12,29,35,17,21)( 8,11,30,36,18,22)$ |
| 6L-1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,26,15, 2,25,16)( 3,27,13, 4,28,14)( 5,23,20,34,32,10)( 6,24,19,33,31, 9)( 7,22,17,36,29,11)( 8,21,18,35,30,12)$ |
| 6M1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,24,15,33,25, 9)( 2,23,16,34,26,10)( 3,22,13,36,28,11)( 4,21,14,35,27,12)( 5,29,20, 7,32,17)( 6,30,19, 8,31,18)$ |
| 6M-1 | $6^{6}$ | $3$ | $6$ | $30$ | $( 1,10,25,34,15,23)( 2, 9,26,33,16,24)( 3,12,28,35,13,21)( 4,11,27,36,14,22)( 5,18,32, 8,20,30)( 6,17,31, 7,19,29)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $72=2^{3} \cdot 3^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 72.48 | magma: IdentifyGroup(G);
| |
| Character table: | 36 x 36 character table |
magma: CharacterTable(G);