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Magma
magma: G := TransitiveGroup(10, 29);
Group action invariants
| Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $((C_2^4 : C_5):C_4)\times C_2$ | ||
| CHM label: | $[2^{5}]F(5)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (5,10), (1,7,9,3)(2,4,8,6), (1,3,5,7,9)(2,4,6,8,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $20$: $F_5$ $40$: $F_{5}\times C_2$ $320$: $(C_2^4 : C_5):C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 5: $F_5$
Low degree siblings
10T29, 20T129, 20T131 x 2, 20T132, 20T133, 20T134, 20T135, 20T137 x 2, 20T140, 32T34608 x 2, 40T460, 40T462, 40T473, 40T474, 40T475, 40T476, 40T487, 40T488, 40T489, 40T490, 40T557, 40T558 x 2, 40T561, 40T562, 40T563, 40T564, 40T565, 40T566, 40T567 x 2, 40T576, 40T577, 40T578, 40T579, 40T586Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,10)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 2, 7)( 5,10)$ |
| $ 2, 2, 2, 1, 1, 1, 1 $ | $10$ | $2$ | $( 2, 7)( 4, 9)( 5,10)$ |
| $ 2, 2, 2, 2, 1, 1 $ | $5$ | $2$ | $( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
| $ 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ |
| $ 5, 5 $ | $64$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ |
| $ 10 $ | $64$ | $10$ | $( 1, 3,10, 2, 4, 6, 8, 5, 7, 9)$ |
| $ 2, 2, 2, 2, 1, 1 $ | $20$ | $2$ | $(1,9)(2,8)(3,7)(4,6)$ |
| $ 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 9)( 2, 8)( 3, 7)( 4, 6)( 5,10)$ |
| $ 4, 2, 2, 1, 1 $ | $40$ | $4$ | $(1,9)(2,8,7,3)(4,6)$ |
| $ 4, 2, 2, 2 $ | $40$ | $4$ | $( 1, 9)( 2, 8, 7, 3)( 4, 6)( 5,10)$ |
| $ 4, 4, 1, 1 $ | $20$ | $4$ | $(1,4,6,9)(2,8,7,3)$ |
| $ 4, 4, 2 $ | $20$ | $4$ | $( 1, 4, 6, 9)( 2, 8, 7, 3)( 5,10)$ |
| $ 4, 4, 1, 1 $ | $40$ | $4$ | $(1,7,9,3)(2,4,8,6)$ |
| $ 4, 4, 2 $ | $40$ | $4$ | $( 1, 7, 9, 3)( 2, 4, 8, 6)( 5,10)$ |
| $ 8, 1, 1 $ | $40$ | $8$ | $(1,2,4,8,6,7,9,3)$ |
| $ 8, 2 $ | $40$ | $8$ | $( 1, 2, 4, 8, 6, 7, 9, 3)( 5,10)$ |
| $ 4, 4, 1, 1 $ | $40$ | $4$ | $(1,3,9,7)(2,6,8,4)$ |
| $ 4, 4, 2 $ | $40$ | $4$ | $( 1, 3, 9, 7)( 2, 6, 8, 4)( 5,10)$ |
| $ 8, 1, 1 $ | $40$ | $8$ | $(1,3,9,2,6,8,4,7)$ |
| $ 8, 2 $ | $40$ | $8$ | $( 1, 3, 9, 2, 6, 8, 4, 7)( 5,10)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $640=2^{7} \cdot 5$ | 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: | 640.21536 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);