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Magma
magma: G := TransitiveGroup(23, 5);
Group action invariants
| Degree $n$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $M_{23}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | 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: | (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (2,14,18,20,8)(3,7,12,13,19)(4,21,17,15,10)(5,11,16,6,9), (3,19)(4,14)(5,20)(6,10)(8,15)(11,18)(17,21)(22,23), (1,22)(2,10)(3,14)(4,17)(8,15)(9,11)(13,20)(19,21) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $3795$ | $2$ | $( 1,14)( 2,16)( 3,21)( 6,15)( 7, 8)(10,12)(13,22)(19,20)$ | |
| $ 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $318780$ | $4$ | $( 1,15,14, 6)( 2,10,16,12)( 3,13,21,22)( 4,17)( 7,20, 8,19)(11,23)$ | |
| $ 8, 8, 4, 2, 1 $ | $1275120$ | $8$ | $( 1, 2,15,10,14,16, 6,12)( 3,19,13, 7,21,20,22, 8)( 4,11,17,23)( 5, 9)$ | |
| $ 11, 11, 1 $ | $927360$ | $11$ | $( 1, 2, 6,14,16,23,15,18, 7, 5, 4)( 3,21,22,10,17, 9,11,20,12,19,13)$ | |
| $ 11, 11, 1 $ | $927360$ | $11$ | $( 1, 4, 5, 7,18,15,23,16,14, 6, 2)( 3,13,19,12,20,11, 9,17,10,22,21)$ | |
| $ 7, 7, 7, 1, 1 $ | $728640$ | $7$ | $( 2, 7,13, 4,20, 5,14)( 3, 6,15,17,11, 9,21)( 8,10,23,22,16,19,12)$ | |
| $ 7, 7, 7, 1, 1 $ | $728640$ | $7$ | $( 2,14, 5,20, 4,13, 7)( 3,21, 9,11,17,15, 6)( 8,12,19,16,22,23,10)$ | |
| $ 14, 7, 2 $ | $728640$ | $14$ | $( 1,18)( 2, 8, 7,10,13,23, 4,22,20,16, 5,19,14,12)( 3,11, 6, 9,15,21,17)$ | |
| $ 14, 7, 2 $ | $728640$ | $14$ | $( 1,18)( 2,10, 4,16,14, 8,13,22, 5,12, 7,23,20,19)( 3, 9,17, 6,21,11,15)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $56672$ | $3$ | $( 1, 7,21)( 2,19,18)( 3,10,22)( 4,16, 5)( 6,13,11)( 8,17,15)$ | |
| $ 6, 6, 3, 3, 2, 2, 1 $ | $850080$ | $6$ | $( 1,21, 7)( 2,16,19, 5,18, 4)( 3,22,10)( 6,17,13,15,11, 8)(12,14)(20,23)$ | |
| $ 5, 5, 5, 5, 1, 1, 1 $ | $680064$ | $5$ | $( 1,17, 5, 6,18)( 2, 7,15, 4,13)( 8,16,11,19,21)( 9,12,23,14,20)$ | |
| $ 15, 5, 3 $ | $680064$ | $15$ | $( 1, 4,19,17,13,21, 5, 2, 8, 6, 7,16,18,15,11)( 3,10,22)( 9,23,20,12,14)$ | |
| $ 15, 5, 3 $ | $680064$ | $15$ | $( 1,16, 2,17,11, 7, 5,19,15, 6,21, 4,18, 8,13)( 3,22,10)( 9,23,20,12,14)$ | |
| $ 23 $ | $443520$ | $23$ | $( 1, 5,12, 3,16,23,11, 2,14,22, 4,19, 6,21,15,17, 8, 9,13,20, 7,18,10)$ | |
| $ 23 $ | $443520$ | $23$ | $( 1,10,18, 7,20,13, 9, 8,17,15,21, 6,19, 4,22,14, 2,11,23,16, 3,12, 5)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $10200960=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 23$ | 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: | 10200960.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| 11 P | |
| 23 P | |
| Type |
magma: CharacterTable(G);