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Magma
magma: G := TransitiveGroup(30, 23);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times F_5$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7,20,13)(2,8,19,14)(3,11,28,29)(4,12,27,30)(5,15)(6,16)(9,24,22,18)(10,23,21,17), (1,21,11,2,22,12)(3,6,20,27,24,25,9,17,13,16,29,8)(4,5,19,28,23,26,10,18,14,15,30,7) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $20$: $F_5$ $24$: $S_3 \times C_4$ $40$: $F_{5}\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: $F_5$
Degree 6: $D_{6}$
Degree 10: $F_{5}\times C_2$
Degree 15: $F_5 \times S_3$
Low degree siblings
15T11, 30T24, 30T32Siblings 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 | $2^{15}$ | $3$ | $2$ | $15$ | $( 1, 2)( 3,23)( 4,24)( 5,16)( 6,15)( 7, 8)( 9,30)(10,29)(11,21)(12,22)(13,14)(17,28)(18,27)(19,20)(25,26)$ |
2B | $2^{12},1^{6}$ | $5$ | $2$ | $12$ | $( 1,20)( 2,19)( 3,28)( 4,27)( 7,13)( 8,14)( 9,22)(10,21)(11,29)(12,30)(17,23)(18,24)$ |
2C | $2^{15}$ | $15$ | $2$ | $15$ | $( 1,19)( 2,20)( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,29)(22,30)(23,28)(24,27)(25,26)$ |
3A | $3^{10}$ | $2$ | $3$ | $20$ | $( 1,11,22)( 2,12,21)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,28)( 8,17,27)( 9,20,29)(10,19,30)$ |
4A1 | $4^{6},2^{3}$ | $5$ | $4$ | $21$ | $( 1, 8,20,14)( 2, 7,19,13)( 3,21,28,10)( 4,22,27, 9)( 5, 6)(11,17,29,23)(12,18,30,24)(15,16)(25,26)$ |
4A-1 | $4^{6},2^{3}$ | $5$ | $4$ | $21$ | $( 1,14,20, 8)( 2,13,19, 7)( 3,10,28,21)( 4, 9,27,22)( 5, 6)(11,23,29,17)(12,24,30,18)(15,16)(25,26)$ |
4B1 | $4^{6},2^{2},1^{2}$ | $15$ | $4$ | $20$ | $( 1, 7,20,13)( 2, 8,19,14)( 3,11,28,29)( 4,12,27,30)( 5,15)( 6,16)( 9,24,22,18)(10,23,21,17)$ |
4B-1 | $4^{6},2^{2},1^{2}$ | $15$ | $4$ | $20$ | $( 1,13,20, 7)( 2,14,19, 8)( 3,29,28,11)( 4,30,27,12)( 5,15)( 6,16)( 9,18,22,24)(10,17,21,23)$ |
5A | $5^{6}$ | $4$ | $5$ | $24$ | $( 1,20, 7,26,13)( 2,19, 8,25,14)( 3,22, 9,28,15)( 4,21,10,27,16)( 5,24,11,29,18)( 6,23,12,30,17)$ |
6A | $6^{4},3^{2}$ | $10$ | $6$ | $24$ | $( 1,24,22,13,11, 3)( 2,23,21,14,12, 4)( 5, 9,26,29,15,20)( 6,10,25,30,16,19)( 7,18,28)( 8,17,27)$ |
10A | $10^{3}$ | $12$ | $10$ | $27$ | $( 1,19, 7,25,13, 2,20, 8,26,14)( 3,12, 9,17,15,23,22,30,28, 6)( 4,11,10,18,16,24,21,29,27, 5)$ |
12A1 | $12^{2},6$ | $10$ | $12$ | $27$ | $( 1,12,22, 2,11,21)( 3, 8,29,16,13,17, 9,25,24,27,20, 6)( 4, 7,30,15,14,18,10,26,23,28,19, 5)$ |
12A-1 | $12^{2},6$ | $10$ | $12$ | $27$ | $( 1,30,15, 8,11,10,26,17,22,19, 5,27)( 2,29,16, 7,12, 9,25,18,21,20, 6,28)( 3,14,24, 4,13,23)$ |
15A | $15^{2}$ | $8$ | $15$ | $28$ | $( 1,18, 3,20, 5,22, 7,24, 9,26,11,28,13,29,15)( 2,17, 4,19, 6,21, 8,23,10,25,12,27,14,30,16)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | 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: | 120.36 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 4A1 | 4A-1 | 4B1 | 4B-1 | 5A | 6A | 10A | 12A1 | 12A-1 | 15A | ||
Size | 1 | 3 | 5 | 15 | 2 | 5 | 5 | 15 | 15 | 4 | 10 | 12 | 10 | 10 | 8 | |
2 P | 1A | 1A | 1A | 1A | 3A | 2B | 2B | 2B | 2B | 5A | 3A | 5A | 6A | 6A | 15A | |
3 P | 1A | 2A | 2B | 2C | 1A | 4A-1 | 4A1 | 4B-1 | 4B1 | 5A | 2B | 10A | 4A1 | 4A-1 | 5A | |
5 P | 1A | 2A | 2B | 2C | 3A | 4A1 | 4A-1 | 4B1 | 4B-1 | 1A | 6A | 2A | 12A1 | 12A-1 | 3A | |
Type | ||||||||||||||||
120.36.1a | R | |||||||||||||||
120.36.1b | R | |||||||||||||||
120.36.1c | R | |||||||||||||||
120.36.1d | R | |||||||||||||||
120.36.1e1 | C | |||||||||||||||
120.36.1e2 | C | |||||||||||||||
120.36.1f1 | C | |||||||||||||||
120.36.1f2 | C | |||||||||||||||
120.36.2a | R | |||||||||||||||
120.36.2b | R | |||||||||||||||
120.36.2c1 | C | |||||||||||||||
120.36.2c2 | C | |||||||||||||||
120.36.4a | R | |||||||||||||||
120.36.4b | R | |||||||||||||||
120.36.8a | R |
magma: CharacterTable(G);