Show commands:
Magma
magma: G := TransitiveGroup(20, 27);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5:F_5$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $10$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,12,18,7)(2,11,17,8)(3,5,15,14)(4,6,16,13)(9,20,10,19), (3,8,12,16,19)(4,7,11,15,20) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: None
Degree 10: $C_5^2 : C_4$
Low degree siblings
10T10 x 2, 20T27, 25T10Siblings 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 | $1^{20}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{10}$ | $25$ | $2$ | $10$ | $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$ |
4A1 | $4^{5}$ | $25$ | $4$ | $15$ | $( 1,12,18, 7)( 2,11,17, 8)( 3, 5,15,14)( 4, 6,16,13)( 9,20,10,19)$ |
4A-1 | $4^{5}$ | $25$ | $4$ | $15$ | $( 1, 7,18,12)( 2, 8,17,11)( 3,14,15, 5)( 4,13,16, 6)( 9,19,10,20)$ |
5A | $5^{4}$ | $4$ | $5$ | $16$ | $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3,12,19, 8,16)( 4,11,20, 7,15)$ |
5B | $5^{4}$ | $4$ | $5$ | $16$ | $( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3, 8,12,16,19)( 4, 7,11,15,20)$ |
5C1 | $5^{4}$ | $4$ | $5$ | $16$ | $( 1,17,14,10, 6)( 2,18,13, 9, 5)( 3,19,16,12, 8)( 4,20,15,11, 7)$ |
5C2 | $5^{2},1^{10}$ | $4$ | $5$ | $8$ | $( 1,10,17, 6,14)( 2, 9,18, 5,13)$ |
5D1 | $5^{2},1^{10}$ | $4$ | $5$ | $8$ | $( 3, 8,12,16,19)( 4, 7,11,15,20)$ |
5D2 | $5^{4}$ | $4$ | $5$ | $16$ | $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Group invariants
Order: | $100=2^{2} \cdot 5^{2}$ | 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: | 100.12 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 4A1 | 4A-1 | 5A | 5B | 5C1 | 5C2 | 5D1 | 5D2 | ||
Size | 1 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 2A | 2A | 5D1 | 5A | 5D2 | 5C2 | 5C1 | 5B | |
5 P | 1A | 2A | 4A1 | 4A-1 | 1A | 1A | 1A | 1A | 1A | 1A | |
Type | |||||||||||
100.12.1a | R | ||||||||||
100.12.1b | R | ||||||||||
100.12.1c1 | C | ||||||||||
100.12.1c2 | C | ||||||||||
100.12.4a | R | ||||||||||
100.12.4b | R | ||||||||||
100.12.4c1 | R | ||||||||||
100.12.4c2 | R | ||||||||||
100.12.4d1 | R | ||||||||||
100.12.4d2 | R |
magma: CharacterTable(G);