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Magma
magma: G := TransitiveGroup(30, 36);
Group action invariants
| Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5\times D_{15}$ | ||
| 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: | (1,9,14,19,27,3,8,13,21,26,2,7,15,20,25)(4,28,22,17,12,6,30,24,16,11,5,29,23,18,10), (1,22,26,16,19,10,15,6,8,29)(2,24,27,18,20,12,13,5,9,28)(3,23,25,17,21,11,14,4,7,30) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $D_{5}$, $C_{10}$ $30$: $D_{15}$, $S_3 \times C_5$ $50$: $D_5\times C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: None
Degree 6: $S_3$
Degree 10: $D_5\times C_5$
Degree 15: None
Low degree siblings
30T36Siblings 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 | $2^{15}$ | $15$ | $2$ | $15$ | $( 1,29)( 2,28)( 3,30)( 4, 7)( 5, 9)( 6, 8)(10,15)(11,14)(12,13)(16,19)(17,21)(18,20)(22,26)(23,25)(24,27)$ |
| 3A | $3^{10}$ | $2$ | $3$ | $20$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$ |
| 5A1 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1,15,26, 8,19)( 2,13,27, 9,20)( 3,14,25, 7,21)( 4,17,30,11,23)( 5,18,28,12,24)( 6,16,29,10,22)$ |
| 5A-1 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1,19, 8,26,15)( 2,20, 9,27,13)( 3,21, 7,25,14)( 4,23,11,30,17)( 5,24,12,28,18)( 6,22,10,29,16)$ |
| 5A2 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1,26,19,15, 8)( 2,27,20,13, 9)( 3,25,21,14, 7)( 4,30,23,17,11)( 5,28,24,18,12)( 6,29,22,16,10)$ |
| 5A-2 | $5^{6}$ | $1$ | $5$ | $24$ | $( 1, 8,15,19,26)( 2, 9,13,20,27)( 3, 7,14,21,25)( 4,11,17,23,30)( 5,12,18,24,28)( 6,10,16,22,29)$ |
| 5B1 | $5^{3},1^{15}$ | $2$ | $5$ | $12$ | $( 1, 8,15,19,26)( 2, 9,13,20,27)( 3, 7,14,21,25)$ |
| 5B2 | $5^{6}$ | $2$ | $5$ | $24$ | $( 1,26,19,15, 8)( 2,27,20,13, 9)( 3,25,21,14, 7)( 4,23,11,30,17)( 5,24,12,28,18)( 6,22,10,29,16)$ |
| 5C1 | $5^{3},1^{15}$ | $2$ | $5$ | $12$ | $( 4,23,11,30,17)( 5,24,12,28,18)( 6,22,10,29,16)$ |
| 5C-1 | $5^{3},1^{15}$ | $2$ | $5$ | $12$ | $( 1,15,26, 8,19)( 2,13,27, 9,20)( 3,14,25, 7,21)$ |
| 5C2 | $5^{6}$ | $2$ | $5$ | $24$ | $( 1,15,26, 8,19)( 2,13,27, 9,20)( 3,14,25, 7,21)( 4,11,17,23,30)( 5,12,18,24,28)( 6,10,16,22,29)$ |
| 5C-2 | $5^{6}$ | $2$ | $5$ | $24$ | $( 1,26,19,15, 8)( 2,27,20,13, 9)( 3,25,21,14, 7)( 4,17,30,11,23)( 5,18,28,12,24)( 6,16,29,10,22)$ |
| 5D1 | $5^{3},1^{15}$ | $2$ | $5$ | $12$ | $( 4,30,23,17,11)( 5,28,24,18,12)( 6,29,22,16,10)$ |
| 5D-1 | $5^{6}$ | $2$ | $5$ | $24$ | $( 1,19, 8,26,15)( 2,20, 9,27,13)( 3,21, 7,25,14)( 4,11,17,23,30)( 5,12,18,24,28)( 6,10,16,22,29)$ |
| 5D2 | $5^{6}$ | $2$ | $5$ | $24$ | $( 1,19, 8,26,15)( 2,20, 9,27,13)( 3,21, 7,25,14)( 4,17,30,11,23)( 5,18,28,12,24)( 6,16,29,10,22)$ |
| 5D-2 | $5^{6}$ | $2$ | $5$ | $24$ | $( 1, 8,15,19,26)( 2, 9,13,20,27)( 3, 7,14,21,25)( 4,30,23,17,11)( 5,28,24,18,12)( 6,29,22,16,10)$ |
| 10A1 | $10^{3}$ | $15$ | $10$ | $27$ | $( 1,22,19,10, 8,29,26,16,15, 6)( 2,24,20,12, 9,28,27,18,13, 5)( 3,23,21,11, 7,30,25,17,14, 4)$ |
| 10A-1 | $10^{3}$ | $15$ | $10$ | $27$ | $( 1, 6,15,16,26,29, 8,10,19,22)( 2, 5,13,18,27,28, 9,12,20,24)( 3, 4,14,17,25,30, 7,11,21,23)$ |
| 10A3 | $10^{3}$ | $15$ | $10$ | $27$ | $( 1,10,26, 6,19,29,15,22, 8,16)( 2,12,27, 5,20,28,13,24, 9,18)( 3,11,25, 4,21,30,14,23, 7,17)$ |
| 10A-3 | $10^{3}$ | $15$ | $10$ | $27$ | $( 1,16, 8,22,15,29,19, 6,26,10)( 2,18, 9,24,13,28,20, 5,27,12)( 3,17, 7,23,14,30,21, 4,25,11)$ |
| 15A1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,27,21,15, 9, 3,26,20,14, 8, 2,25,19,13, 7)( 4,18,29,11,24, 6,17,28,10,23, 5,16,30,12,22)$ |
| 15A-1 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1, 9,14,19,27, 3, 8,13,21,26, 2, 7,15,20,25)( 4, 5, 6)(10,11,12)(16,17,18)(22,23,24)(28,29,30)$ |
| 15A2 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,27,21,15, 9, 3,26,20,14, 8, 2,25,19,13, 7)( 4,24,10,30,18, 6,23,12,29,17, 5,22,11,28,16)$ |
| 15A-2 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1, 2, 3)( 4,24,10,30,18, 6,23,12,29,17, 5,22,11,28,16)( 7, 8, 9)(13,14,15)(19,20,21)(25,26,27)$ |
| 15B1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,20, 7,26,13, 3,19, 9,25,15, 2,21, 8,27,14)( 4,28,22,17,12, 6,30,24,16,11, 5,29,23,18,10)$ |
| 15B2 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,13,25, 8,20, 3,15,27, 7,19, 2,14,26, 9,21)( 4,24,10,30,18, 6,23,12,29,17, 5,22,11,28,16)$ |
| 15B4 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1, 9,14,19,27, 3, 8,13,21,26, 2, 7,15,20,25)( 4,18,29,11,24, 6,17,28,10,23, 5,16,30,12,22)$ |
| 15B7 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1, 2, 3)( 4,18,29,11,24, 6,17,28,10,23, 5,16,30,12,22)( 7, 8, 9)(13,14,15)(19,20,21)(25,26,27)$ |
| 15C1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,13,25, 8,20, 3,15,27, 7,19, 2,14,26, 9,21)( 4,28,22,17,12, 6,30,24,16,11, 5,29,23,18,10)$ |
| 15C-1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1, 9,14,19,27, 3, 8,13,21,26, 2, 7,15,20,25)( 4,28,22,17,12, 6,30,24,16,11, 5,29,23,18,10)$ |
| 15C2 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,20, 7,26,13, 3,19, 9,25,15, 2,21, 8,27,14)( 4,18,29,11,24, 6,17,28,10,23, 5,16,30,12,22)$ |
| 15C-2 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1,27,21,15, 9, 3,26,20,14, 8, 2,25,19,13, 7)( 4, 5, 6)(10,11,12)(16,17,18)(22,23,24)(28,29,30)$ |
| 15C4 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1, 2, 3)( 4,12,16,23,28, 6,11,18,22,30, 5,10,17,24,29)( 7, 8, 9)(13,14,15)(19,20,21)(25,26,27)$ |
| 15C-4 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,13,25, 8,20, 3,15,27, 7,19, 2,14,26, 9,21)( 4,12,16,23,28, 6,11,18,22,30, 5,10,17,24,29)$ |
| 15C7 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1, 9,14,19,27, 3, 8,13,21,26, 2, 7,15,20,25)( 4,12,16,23,28, 6,11,18,22,30, 5,10,17,24,29)$ |
| 15C-7 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,27,21,15, 9, 3,26,20,14, 8, 2,25,19,13, 7)( 4,12,16,23,28, 6,11,18,22,30, 5,10,17,24,29)$ |
| 15D1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,13,25, 8,20, 3,15,27, 7,19, 2,14,26, 9,21)( 4,18,29,11,24, 6,17,28,10,23, 5,16,30,12,22)$ |
| 15D-1 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,20, 7,26,13, 3,19, 9,25,15, 2,21, 8,27,14)( 4,24,10,30,18, 6,23,12,29,17, 5,22,11,28,16)$ |
| 15D2 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,27,21,15, 9, 3,26,20,14, 8, 2,25,19,13, 7)( 4,28,22,17,12, 6,30,24,16,11, 5,29,23,18,10)$ |
| 15D-2 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1,20, 7,26,13, 3,19, 9,25,15, 2,21, 8,27,14)( 4, 5, 6)(10,11,12)(16,17,18)(22,23,24)(28,29,30)$ |
| 15D4 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1,20, 7,26,13, 3,19, 9,25,15, 2,21, 8,27,14)( 4,12,16,23,28, 6,11,18,22,30, 5,10,17,24,29)$ |
| 15D-4 | $15^{2}$ | $2$ | $15$ | $28$ | $( 1, 9,14,19,27, 3, 8,13,21,26, 2, 7,15,20,25)( 4,24,10,30,18, 6,23,12,29,17, 5,22,11,28,16)$ |
| 15D7 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1, 2, 3)( 4,28,22,17,12, 6,30,24,16,11, 5,29,23,18,10)( 7, 8, 9)(13,14,15)(19,20,21)(25,26,27)$ |
| 15D-7 | $15,3^{5}$ | $2$ | $15$ | $24$ | $( 1,13,25, 8,20, 3,15,27, 7,19, 2,14,26, 9,21)( 4, 5, 6)(10,11,12)(16,17,18)(22,23,24)(28,29,30)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $150=2 \cdot 3 \cdot 5^{2}$ | 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: | 150.11 | magma: IdentifyGroup(G);
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| Character table: | 45 x 45 character table |
magma: CharacterTable(G);