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Magma
magma: G := TransitiveGroup(30, 15);
Group action invariants
| Degree $n$: | $30$ | 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: | $S_3\times C_{15}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $15$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29), (1,30,13,26,11,22,7,19,5,18,3,29,15,25,10,24,9,21,4,17,2,28,14,27,12,23,8,20,6,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $5$: $C_5$ $6$: $S_3$, $C_6$ $10$: $C_{10}$ $15$: $C_{15}$ $18$: $S_3\times C_3$ $30$: $S_3 \times C_5$, $C_{30}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 5: $C_5$
Degree 6: $S_3\times C_3$
Degree 10: $C_{10}$
Degree 15: None
Low degree siblings
45T3Siblings 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 | $1^{30}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{15}$ | $3$ | $2$ | $15$ | $( 1,22)( 2,23)( 3,24)( 4,27)( 5,25)( 6,26)( 7,29)( 8,30)( 9,28)(10,17)(11,18)(12,16)(13,19)(14,20)(15,21)$ |
| 3A1 | $3^{10}$ | $1$ | $3$ | $20$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$ |
| 3A-1 | $3^{10}$ | $1$ | $3$ | $20$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$ |
| 3B | $3^{5},1^{15}$ | $2$ | $3$ | $10$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$ |
| 3C1 | $3^{10}$ | $2$ | $3$ | $20$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$ |
| 3C-1 | $3^{5},1^{15}$ | $2$ | $3$ | $10$ | $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$ |
| 5A1 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1,15,12, 7, 4)( 2,13,10, 8, 5)( 3,14,11, 9, 6)(16,29,27,22,21)(17,30,25,23,19)(18,28,26,24,20)$ |
| 5A-1 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1,12, 4,15, 7)( 2,10, 5,13, 8)( 3,11, 6,14, 9)(16,27,21,29,22)(17,25,19,30,23)(18,26,20,28,24)$ |
| 5A2 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1, 4, 7,12,15)( 2, 5, 8,10,13)( 3, 6, 9,11,14)(16,21,22,27,29)(17,19,23,25,30)(18,20,24,26,28)$ |
| 5A-2 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1, 7,15, 4,12)( 2, 8,13, 5,10)( 3, 9,14, 6,11)(16,22,29,21,27)(17,23,30,19,25)(18,24,28,20,26)$ |
| 6A1 | $6^{5}$ | $3$ | $6$ | $25$ | $( 1,23, 3,22, 2,24)( 4,25, 6,27, 5,26)( 7,30, 9,29, 8,28)(10,18,12,17,11,16)(13,20,15,19,14,21)$ |
| 6A-1 | $6^{5}$ | $3$ | $6$ | $25$ | $( 1,24, 2,22, 3,23)( 4,26, 5,27, 6,25)( 7,28, 8,29, 9,30)(10,16,11,17,12,18)(13,21,14,19,15,20)$ |
| 10A1 | $10^{3}$ | $3$ | $10$ | $27$ | $( 1,16, 4,21, 7,22,12,27,15,29)( 2,17, 5,19, 8,23,10,25,13,30)( 3,18, 6,20, 9,24,11,26,14,28)$ |
| 10A-1 | $10^{3}$ | $3$ | $10$ | $27$ | $( 1,21,12,29, 4,22,15,16, 7,27)( 2,19,10,30, 5,23,13,17, 8,25)( 3,20,11,28, 6,24,14,18, 9,26)$ |
| 10A3 | $10^{3}$ | $3$ | $10$ | $27$ | $( 1,29,15,27,12,22, 7,21, 4,16)( 2,30,13,25,10,23, 8,19, 5,17)( 3,28,14,26,11,24, 9,20, 6,18)$ |
| 10A-3 | $10^{3}$ | $3$ | $10$ | $27$ | $( 1,27, 7,16,15,22, 4,29,12,21)( 2,25, 8,17,13,23, 5,30,10,19)( 3,26, 9,18,14,24, 6,28,11,20)$ |
| 15A1 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1, 6, 8,12,14, 2, 4, 9,10,15, 3, 5, 7,11,13)(16,20,23,27,28,17,21,24,25,29,18,19,22,26,30)$ |
| 15A-1 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1, 9,13, 4,11, 2, 7,14, 5,12, 3, 8,15, 6,10)(16,24,30,21,26,17,22,28,19,27,18,23,29,20,25)$ |
| 15A2 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1,14,10, 7, 6, 2,15,11, 8, 4, 3,13,12, 9, 5)(16,28,25,22,20,17,29,26,23,21,18,30,27,24,19)$ |
| 15A-2 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1,10, 6,15, 8, 3,12, 5,14, 7, 2,11, 4,13, 9)(16,25,20,29,23,18,27,19,28,22,17,26,21,30,24)$ |
| 15A4 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1, 5, 9,12,13, 3, 4, 8,11,15, 2, 6, 7,10,14)(16,19,24,27,30,18,21,23,26,29,17,20,22,25,28)$ |
| 15A-4 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1, 8,14, 4,10, 3, 7,13, 6,12, 2, 9,15, 5,11)(16,23,28,21,25,18,22,30,20,27,17,24,29,19,26)$ |
| 15A7 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1,13,11, 7, 5, 3,15,10, 9, 4, 2,14,12, 8, 6)(16,30,26,22,19,18,29,25,24,21,17,28,27,23,20)$ |
| 15A-7 | $15^{2}$ | $1$ | $15$ | $28$ | $( 1,11, 5,15, 9, 2,12, 6,13, 7, 3,10, 4,14, 8)(16,26,19,29,24,17,27,20,30,22,18,25,21,28,23)$ |
| 15B1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1, 5, 9,12,13, 3, 4, 8,11,15, 2, 6, 7,10,14)(16,20,23,27,28,17,21,24,25,29,18,19,22,26,30)$ |
| 15B-1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,10, 6,15, 8, 3,12, 5,14, 7, 2,11, 4,13, 9)(16,26,19,29,24,17,27,20,30,22,18,25,21,28,23)$ |
| 15B2 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1, 9,13, 4,11, 2, 7,14, 5,12, 3, 8,15, 6,10)(16,22,29,21,27)(17,23,30,19,25)(18,24,28,20,26)$ |
| 15B-2 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1,14,10, 7, 6, 2,15,11, 8, 4, 3,13,12, 9, 5)(16,29,27,22,21)(17,30,25,23,19)(18,28,26,24,20)$ |
| 15C1 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1, 6, 8,12,14, 2, 4, 9,10,15, 3, 5, 7,11,13)(16,21,22,27,29)(17,19,23,25,30)(18,20,24,26,28)$ |
| 15C-1 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1,12, 4,15, 7)( 2,10, 5,13, 8)( 3,11, 6,14, 9)(16,25,20,29,23,18,27,19,28,22,17,26,21,30,24)$ |
| 15C2 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,13,11, 7, 5, 3,15,10, 9, 4, 2,14,12, 8, 6)(16,28,25,22,20,17,29,26,23,21,18,30,27,24,19)$ |
| 15C-2 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1,15,12, 7, 4)( 2,13,10, 8, 5)( 3,14,11, 9, 6)(16,30,26,22,19,18,29,25,24,21,17,28,27,23,20)$ |
| 15C4 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1,11, 5,15, 9, 2,12, 6,13, 7, 3,10, 4,14, 8)(16,27,21,29,22)(17,25,19,30,23)(18,26,20,28,24)$ |
| 15C-4 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1, 4, 7,12,15)( 2, 5, 8,10,13)( 3, 6, 9,11,14)(16,19,24,27,30,18,21,23,26,29,17,20,22,25,28)$ |
| 15C7 | $15,5^{3}$ | $2$ | $15$ | $26$ | $( 1, 7,15, 4,12)( 2, 8,13, 5,10)( 3, 9,14, 6,11)(16,23,28,21,25,18,22,30,20,27,17,24,29,19,26)$ |
| 15C-7 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1, 8,14, 4,10, 3, 7,13, 6,12, 2, 9,15, 5,11)(16,24,30,21,26,17,22,28,19,27,18,23,29,20,25)$ |
| 30A1 | $30$ | $3$ | $30$ | $29$ | $( 1,25, 9,16,13,24, 4,30,11,21, 2,26, 7,17,14,22, 5,28,12,19, 3,27, 8,18,15,23, 6,29,10,20)$ |
| 30A-1 | $30$ | $3$ | $30$ | $29$ | $( 1,19,11,29, 5,24,15,17, 9,27, 2,20,12,30, 6,22,13,18, 7,25, 3,21,10,28, 4,23,14,16, 8,26)$ |
| 30A7 | $30$ | $3$ | $30$ | $29$ | $( 1,26, 8,16,14,23, 4,28,10,21, 3,25, 7,18,13,22, 6,30,12,20, 2,27, 9,17,15,24, 5,29,11,19)$ |
| 30A-7 | $30$ | $3$ | $30$ | $29$ | $( 1,28,13,27,11,23, 7,20, 5,16, 3,30,15,26,10,22, 9,19, 4,18, 2,29,14,25,12,24, 8,21, 6,17)$ |
| 30A11 | $30$ | $3$ | $30$ | $29$ | $( 1,30,14,27,10,24, 7,19, 6,16, 2,28,15,25,11,22, 8,20, 4,17, 3,29,13,26,12,23, 9,21, 5,18)$ |
| 30A-11 | $30$ | $3$ | $30$ | $29$ | $( 1,17, 6,21, 8,24,12,25,14,29, 2,18, 4,19, 9,22,10,26,15,30, 3,16, 5,20, 7,23,11,27,13,28)$ |
| 30A13 | $30$ | $3$ | $30$ | $29$ | $( 1,20,10,29, 6,23,15,18, 8,27, 3,19,12,28, 5,22,14,17, 7,26, 2,21,11,30, 4,24,13,16, 9,25)$ |
| 30A-13 | $30$ | $3$ | $30$ | $29$ | $( 1,18, 5,21, 9,23,12,26,13,29, 3,17, 4,20, 8,22,11,25,15,28, 2,16, 6,19, 7,24,10,27,14,30)$ |
magma: ConjugacyClasses(G);
Malle's constant $a(G)$: $1/10$
Group invariants
| Order: | $90=2 \cdot 3^{2} \cdot 5$ | 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: | 90.6 | magma: IdentifyGroup(G);
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| Character table: | 45 x 45 character table |
magma: CharacterTable(G);