Show commands:
Magma
magma: G := TransitiveGroup(10, 43);
Group action invariants
| Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_5^2 \wr C_2$ | ||
| CHM label: | $[S(5)^{2}]2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: None
Low degree siblings
12T288, 20T539, 20T540, 20T542, 20T544, 24T13996, 24T13997, 24T13998, 25T106, 30T1011, 36T13308, 40T14374, 40T14375, 40T14376, 40T14377, 40T14378, 40T14379, 40T14380, 40T14381Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $100$ | $2$ | $(1,3)(6,8)$ | |
| $ 2, 2, 2, 2, 1, 1 $ | $225$ | $2$ | $( 1, 3)( 2,10)( 5, 7)( 6, 8)$ | |
| $ 3, 3, 1, 1, 1, 1 $ | $400$ | $3$ | $( 1, 3, 5)( 6, 8,10)$ | |
| $ 3, 3, 2, 2 $ | $400$ | $6$ | $( 1, 3, 5)( 2, 4)( 6, 8,10)( 7, 9)$ | |
| $ 4, 4, 1, 1 $ | $900$ | $4$ | $( 1, 3, 5, 7)( 2, 6, 8,10)$ | |
| $ 5, 5 $ | $576$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ | |
| $ 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $2$ | $(6,8)$ | |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $30$ | $2$ | $( 2,10)( 6, 8)$ | |
| $ 3, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $( 6, 8,10)$ | |
| $ 3, 2, 1, 1, 1, 1, 1 $ | $40$ | $6$ | $( 2, 4)( 6, 8,10)$ | |
| $ 4, 1, 1, 1, 1, 1, 1 $ | $60$ | $4$ | $( 2, 6, 8,10)$ | |
| $ 5, 1, 1, 1, 1, 1 $ | $48$ | $5$ | $( 2, 4, 6, 8,10)$ | |
| $ 2, 2, 2, 1, 1, 1, 1 $ | $300$ | $2$ | $( 1, 3)( 2,10)( 6, 8)$ | |
| $ 3, 2, 1, 1, 1, 1, 1 $ | $400$ | $6$ | $( 1, 3)( 6, 8,10)$ | |
| $ 3, 2, 2, 1, 1, 1 $ | $400$ | $6$ | $( 1, 3)( 2, 4)( 6, 8,10)$ | |
| $ 4, 2, 1, 1, 1, 1 $ | $600$ | $4$ | $( 1, 3)( 2, 6, 8,10)$ | |
| $ 5, 2, 1, 1, 1 $ | $480$ | $10$ | $( 1, 3)( 2, 4, 6, 8,10)$ | |
| $ 3, 2, 2, 1, 1, 1 $ | $600$ | $6$ | $( 1, 3)( 5, 7)( 6, 8,10)$ | |
| $ 3, 2, 2, 2, 1 $ | $600$ | $6$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8,10)$ | |
| $ 4, 2, 2, 1, 1 $ | $900$ | $4$ | $( 1, 3)( 2, 6, 8,10)( 5, 7)$ | |
| $ 5, 2, 2, 1 $ | $720$ | $10$ | $( 1, 3)( 2, 4, 6, 8,10)( 5, 7)$ | |
| $ 3, 3, 2, 1, 1 $ | $800$ | $6$ | $( 1, 3, 5)( 2, 4)( 6, 8,10)$ | |
| $ 4, 3, 1, 1, 1 $ | $1200$ | $12$ | $( 1, 3, 5)( 2, 6, 8,10)$ | |
| $ 5, 3, 1, 1 $ | $960$ | $15$ | $( 1, 3, 5)( 2, 4, 6, 8,10)$ | |
| $ 4, 3, 2, 1 $ | $1200$ | $12$ | $( 1, 3, 5)( 2, 6, 8,10)( 7, 9)$ | |
| $ 5, 3, 2 $ | $960$ | $30$ | $( 1, 3, 5)( 2, 4, 6, 8,10)( 7, 9)$ | |
| $ 5, 4, 1 $ | $1440$ | $20$ | $( 1, 3, 5, 7)( 2, 4, 6, 8,10)$ | |
| $ 2, 2, 2, 2, 2 $ | $120$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ | |
| $ 4, 2, 2, 2 $ | $1200$ | $4$ | $( 1, 8, 3, 6)( 2, 7)( 4, 9)( 5,10)$ | |
| $ 4, 4, 2 $ | $1800$ | $4$ | $( 1, 8, 3, 6)( 2, 7,10, 5)( 4, 9)$ | |
| $ 6, 2, 2 $ | $2400$ | $6$ | $( 1, 8, 3,10, 5, 6)( 2, 7)( 4, 9)$ | |
| $ 6, 4 $ | $2400$ | $12$ | $( 1, 8, 3,10, 5, 6)( 2, 7, 4, 9)$ | |
| $ 10 $ | $2880$ | $10$ | $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$ | |
| $ 8, 2 $ | $3600$ | $8$ | $( 1, 8, 3,10, 5, 2, 7, 6)( 4, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $28800=2^{7} \cdot 3^{2} \cdot 5^{2}$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 28800.r | magma: IdentifyGroup(G);
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| Character table: | 35 x 35 character table |
magma: CharacterTable(G);