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Magma
magma: G := TransitiveGroup(30, 30);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\times A_5$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,21,26,20,3,2,22,25,19,4)(5,27,29,7,11,6,28,30,8,12)(9,17,16,13,24,10,18,15,14,23), (1,25,5,2,26,6)(3,18,8,4,17,7)(9,19,30,10,20,29)(11,27,13,12,28,14)(15,24,22,16,23,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 5: $A_5$
Degree 6: None
Degree 10: $A_5\times C_2$
Degree 15: $A_5$
Low degree siblings
10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 40T61Siblings 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 |
$1^{30}$ | $1$ | $1$ | $()$ | |
$2^{12},1^{6}$ | $15$ | $2$ | $( 3,10)( 4, 9)( 5,26)( 6,25)( 7,20)( 8,19)(13,28)(14,27)(17,29)(18,30)(21,24) (22,23)$ | |
$2^{15}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$ | |
$2^{15}$ | $15$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5,25)( 6,26)( 7,19)( 8,20)(11,12)(13,27)(14,28)(15,16) (17,30)(18,29)(21,23)(22,24)$ | |
$5^{6}$ | $12$ | $5$ | $( 1, 3,13,28,10)( 2, 4,14,27, 9)( 5,26,29,15,17)( 6,25,30,16,18) ( 7,21,24,20,12)( 8,22,23,19,11)$ | |
$5^{6}$ | $12$ | $5$ | $( 1, 3,19,26,22)( 2, 4,20,25,21)( 5,11, 8,29,28)( 6,12, 7,30,27) ( 9,24,14,16,18)(10,23,13,15,17)$ | |
$3^{10}$ | $20$ | $3$ | $( 1, 3,23)( 2, 4,24)( 5,22,13)( 6,21,14)( 7,25,12)( 8,26,11)( 9,20,30) (10,19,29)(15,17,28)(16,18,27)$ | |
$10^{3}$ | $12$ | $10$ | $( 1, 4,13,27,10, 2, 3,14,28, 9)( 5,25,29,16,17, 6,26,30,15,18)( 7,22,24,19,12, 8,21,23,20,11)$ | |
$10^{3}$ | $12$ | $10$ | $( 1, 4,19,25,22, 2, 3,20,26,21)( 5,12, 8,30,28, 6,11, 7,29,27)( 9,23,14,15,18, 10,24,13,16,17)$ | |
$6^{5}$ | $20$ | $6$ | $( 1, 4,23, 2, 3,24)( 5,21,13, 6,22,14)( 7,26,12, 8,25,11)( 9,19,30,10,20,29) (15,18,28,16,17,27)$ |
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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 120.35 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 5A1 | 5A2 | 6A | 10A1 | 10A3 | ||
Size | 1 | 1 | 15 | 15 | 20 | 12 | 12 | 20 | 12 | 12 | |
2 P | 1A | 1A | 1A | 1A | 3A | 5A2 | 5A1 | 3A | 5A1 | 5A2 | |
3 P | 1A | 2A | 2B | 2C | 1A | 5A2 | 5A1 | 2A | 10A3 | 10A1 | |
5 P | 1A | 2A | 2B | 2C | 3A | 1A | 1A | 6A | 2A | 2A | |
Type | |||||||||||
120.35.1a | R | ||||||||||
120.35.1b | R | ||||||||||
120.35.3a1 | R | ||||||||||
120.35.3a2 | R | ||||||||||
120.35.3b1 | R | ||||||||||
120.35.3b2 | R | ||||||||||
120.35.4a | R | ||||||||||
120.35.4b | R | ||||||||||
120.35.5a | R | ||||||||||
120.35.5b | R |
magma: CharacterTable(G);