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Magma
magma: G := TransitiveGroup(30, 40);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:C_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $10$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,19,7,25,14)(2,20,8,26,13)(5,17,29,11,23)(6,18,30,12,24), (1,16,5,2,15,6)(3,24,19,4,23,20)(7,22,11,8,21,12)(9,30,25,10,29,26)(13,27,18,14,28,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $75$: $C_5^2 : C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 5: None
Degree 6: $C_6$
Degree 10: None
Degree 15: $C_5^2 : C_3$
Low degree siblings
30T40Siblings 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 | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 3, 9,15,21,27)( 4,10,16,22,28)( 5,29,23,17,11)( 6,30,24,18,12)$ | |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 3,15,27, 9,21)( 4,16,28,10,22)( 5,23,11,29,17)( 6,24,12,30,18)$ | |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 3,21, 9,27,15)( 4,22,10,28,16)( 5,17,29,11,23)( 6,18,30,12,24)$ | |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 3,27,21,15, 9)( 4,28,22,16,10)( 5,11,17,23,29)( 6,12,18,24,30)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$ | |
$ 10, 10, 2, 2, 2, 2, 2 $ | $3$ | $10$ | $( 1, 2)( 3,10,15,22,27, 4, 9,16,21,28)( 5,30,23,18,11, 6,29,24,17,12)( 7, 8) (13,14)(19,20)(25,26)$ | |
$ 10, 10, 2, 2, 2, 2, 2 $ | $3$ | $10$ | $( 1, 2)( 3,16,27,10,21, 4,15,28, 9,22)( 5,24,11,30,17, 6,23,12,29,18)( 7, 8) (13,14)(19,20)(25,26)$ | |
$ 10, 10, 2, 2, 2, 2, 2 $ | $3$ | $10$ | $( 1, 2)( 3,22, 9,28,15, 4,21,10,27,16)( 5,18,29,12,23, 6,17,30,11,24)( 7, 8) (13,14)(19,20)(25,26)$ | |
$ 10, 10, 2, 2, 2, 2, 2 $ | $3$ | $10$ | $( 1, 2)( 3,28,21,16, 9, 4,27,22,15,10)( 5,12,17,24,29, 6,11,18,23,30)( 7, 8) (13,14)(19,20)(25,26)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,16,18)(14,15,17)(19,21,23) (20,22,24)(25,27,29)(26,28,30)$ | |
$ 6, 6, 6, 6, 6 $ | $25$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,10,11, 8, 9,12)(13,15,18,14,16,17)(19,22,23,20,21,24) (25,28,29,26,27,30)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)(13,18,16)(14,17,15)(19,23,21) (20,24,22)(25,29,27)(26,30,28)$ | |
$ 6, 6, 6, 6, 6 $ | $25$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,12, 9, 8,11,10)(13,17,16,14,18,15)(19,24,21,20,23,22) (25,30,27,26,29,28)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,15,21,27)( 4,10,16,22,28) ( 5,23,11,29,17)( 6,24,12,30,18)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3,15,27, 9,21)( 4,16,28,10,22) ( 5,17,29,11,23)( 6,18,30,12,24)$ | |
$ 10, 10, 10 $ | $3$ | $10$ | $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,15,22,27, 4, 9,16,21,28)( 5,24,11,30,17, 6,23,12,29,18)$ | |
$ 10, 10, 10 $ | $3$ | $10$ | $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,16,27,10,21, 4,15,28, 9,22)( 5,18,29,12,23, 6,17,30,11,24)$ | |
$ 10, 10, 10 $ | $3$ | $10$ | $( 1,13,25, 8,19, 2,14,26, 7,20)( 3,28,21,16, 9, 4,27,22,15,10)( 5,30,23,18,11, 6,29,24,17,12)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1,14,25, 7,19)( 2,13,26, 8,20)( 3,27,21,15, 9)( 4,28,22,16,10) ( 5,29,23,17,11)( 6,30,24,18,12)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1,19, 7,25,14)( 2,20, 8,26,13)( 3,21, 9,27,15)( 4,22,10,28,16) ( 5,29,23,17,11)( 6,30,24,18,12)$ | |
$ 10, 10, 10 $ | $3$ | $10$ | $( 1,20, 7,26,14, 2,19, 8,25,13)( 3,22, 9,28,15, 4,21,10,27,16)( 5,30,23,18,11, 6,29,24,17,12)$ |
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: | 150.7 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 5A1 | 5A-1 | 5A2 | 5A-2 | 5B1 | 5B-1 | 5B2 | 5B-2 | 6A1 | 6A-1 | 10A1 | 10A-1 | 10A3 | 10A-3 | 10B1 | 10B-1 | 10B3 | 10B-3 | ||
Size | 1 | 1 | 25 | 25 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 25 | 25 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 5B1 | 5A1 | 5A2 | 5B-1 | 5A-2 | 5B2 | 5B-2 | 5A-1 | 3A1 | 3A-1 | 5A1 | 5B-2 | 5A2 | 5B2 | 5A-1 | 5B-1 | 5B1 | 5A-2 | |
3 P | 1A | 2A | 1A | 1A | 5B-1 | 5A-1 | 5A-2 | 5B1 | 5A2 | 5B-2 | 5B2 | 5A1 | 2A | 2A | 10A3 | 10B-1 | 10A1 | 10B1 | 10A-3 | 10B-3 | 10B3 | 10A-1 | |
5 P | 1A | 2A | 3A-1 | 3A1 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 6A-1 | 6A1 | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||||||||||||
150.7.1a | R | ||||||||||||||||||||||
150.7.1b | R | ||||||||||||||||||||||
150.7.1c1 | C | ||||||||||||||||||||||
150.7.1c2 | C | ||||||||||||||||||||||
150.7.1d1 | C | ||||||||||||||||||||||
150.7.1d2 | C | ||||||||||||||||||||||
150.7.3a1 | C | ||||||||||||||||||||||
150.7.3a2 | C | ||||||||||||||||||||||
150.7.3a3 | C | ||||||||||||||||||||||
150.7.3a4 | C | ||||||||||||||||||||||
150.7.3b1 | C | ||||||||||||||||||||||
150.7.3b2 | C | ||||||||||||||||||||||
150.7.3b3 | C | ||||||||||||||||||||||
150.7.3b4 | C | ||||||||||||||||||||||
150.7.3c1 | C | ||||||||||||||||||||||
150.7.3c2 | C | ||||||||||||||||||||||
150.7.3c3 | C | ||||||||||||||||||||||
150.7.3c4 | C | ||||||||||||||||||||||
150.7.3d1 | C | ||||||||||||||||||||||
150.7.3d2 | C | ||||||||||||||||||||||
150.7.3d3 | C | ||||||||||||||||||||||
150.7.3d4 | C |
magma: CharacterTable(G);