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Magma
magma: G := TransitiveGroup(40, 14344);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $14344$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\PSp(4,3)$ | ||
| 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: | (1,3,8,16,7)(2,6,13,19,17)(4,9,5,10,20)(11,23,36,26,35)(12,21,31,25,18)(14,27,29,22,34)(15,28,30,38,32)(24,37,40,39,33), (1,2,5,10,19)(3,4,7,15,28)(6,12,21,13,25)(8,16,29,38,14)(9,18,31,20,27)(11,22,33,37,34)(17,30,23,35,32)(24,26,39,40,36) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: None
Degree 8: None
Degree 10: None
Degree 20: None
Low degree siblings
27T993, 36T12781, 40T14345, 45T666Siblings 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 | $1^{40}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{16},1^{8}$ | $45$ | $2$ | $16$ | $( 2, 9)( 3,29)( 4,38)( 5,15)( 6,27)( 7,28)(10,31)(11,40)(14,35)(16,30)(17,22)(18,23)(20,21)(24,36)(26,34)(32,37)$ |
| 2B | $2^{20}$ | $270$ | $2$ | $20$ | $( 1,39)( 2,22)( 3,24)( 4,28)( 5,20)( 6,23)( 7,38)( 8,25)( 9,17)(10,14)(11,32)(12,19)(13,33)(15,21)(16,26)(18,27)(29,36)(30,34)(31,35)(37,40)$ |
| 3A1 | $3^{9},1^{13}$ | $40$ | $3$ | $18$ | $( 2,39,20)( 6, 8,30)( 9,40,19)(11,21,12)(13,32,16)(14,28,31)(15,29,18)(17,26,36)(27,37,33)$ |
| 3A-1 | $3^{9},1^{13}$ | $40$ | $3$ | $18$ | $( 2,20,39)( 6,30, 8)( 9,19,40)(11,12,21)(13,16,32)(14,31,28)(15,18,29)(17,36,26)(27,33,37)$ |
| 3B | $3^{12},1^{4}$ | $240$ | $3$ | $24$ | $( 1, 5,22)( 2,39,20)( 4,35,10)( 6, 8,30)( 9,13,15)(11,36,37)(12,26,27)(14,28,31)(16,18,19)(17,33,21)(23,25,34)(29,40,32)$ |
| 3C | $3^{12},1^{4}$ | $480$ | $3$ | $24$ | $( 1,25, 4)( 2,28,30)( 5,34,35)( 6,39,31)( 8,20,14)( 9,29,16)(10,22,23)(11,17,27)(12,36,33)(13,40,18)(15,32,19)(21,26,37)$ |
| 4A | $4^{8},2^{2},1^{4}$ | $540$ | $4$ | $26$ | $( 2, 5, 9,15)( 3,14,29,35)( 4,37,38,32)( 6,30,27,16)( 7,11,28,40)( 8,25)(10,36,31,24)(13,33)(17,21,22,20)(18,26,23,34)$ |
| 4B | $4^{10}$ | $3240$ | $4$ | $30$ | $( 1,13,39,33)( 2,20,22, 5)( 3,28,24, 4)( 6,34,23,30)( 7,36,38,29)( 8,12,25,19)( 9,21,17,15)(10,11,14,32)(16,27,26,18)(31,40,35,37)$ |
| 5A | $5^{8}$ | $5184$ | $5$ | $32$ | $( 1,33,14,26,37)( 2,13,23, 6,39)( 3,20, 5,28,17)( 4,24,32,30,35)( 7,34,29,27,18)( 8,16,19,15,21)( 9,31,38,36,11)(10,22,40,25,12)$ |
| 6A1 | $6^{4},3,2^{4},1^{5}$ | $360$ | $6$ | $26$ | $( 1, 4)( 2,20,39)( 3,24)( 5,35)( 6,31, 8,14,30,28)( 9,21,40,12,19,11)(10,22)(13,17,32,26,16,36)(15,33,29,27,18,37)$ |
| 6A-1 | $6^{4},3,2^{4},1^{5}$ | $360$ | $6$ | $26$ | $( 1, 4)( 2,39,20)( 3,24)( 5,35)( 6,28,30,14, 8,31)( 9,11,19,12,40,21)(10,22)(13,36,16,26,32,17)(15,37,18,27,29,33)$ |
| 6B1 | $6^{5},3^{2},2,1^{2}$ | $720$ | $6$ | $30$ | $( 1, 6, 5, 8,22,30)( 2,34,39,23,20,25)( 4,14,35,28,10,31)( 7,38)( 9,16,13,18,15,19)(11,37,36)(12,17,26,33,27,21)(29,32,40)$ |
| 6B-1 | $6^{5},3^{2},2,1^{2}$ | $720$ | $6$ | $30$ | $( 1,30,22, 8, 5, 6)( 2,25,20,23,39,34)( 4,31,10,28,35,14)( 7,38)( 9,19,15,18,13,16)(11,36,37)(12,21,27,33,26,17)(29,40,32)$ |
| 6C | $6^{5},3^{2},2,1^{2}$ | $1440$ | $6$ | $30$ | $( 1,36,25,33, 4,12)( 2,30,28)( 5,27,34,11,35,17)( 6,20,39,14,31, 8)( 9,16,29)(10,37,22,21,23,26)(13,32,40,19,18,15)(24,38)$ |
| 6D | $6^{6},2^{2}$ | $2160$ | $6$ | $32$ | $( 1, 2, 5,39,22,20)( 3,24)( 4,14,35,28,10,31)( 6,34, 8,23,30,25)( 7,38)( 9,21,13,17,15,33)(11,40,36,32,37,29)(12,18,26,19,27,16)$ |
| 9A1 | $9^{3},3^{4},1$ | $2880$ | $9$ | $32$ | $( 1,23,34)( 2,13,37,39,32,33,20,16,27)( 3,24, 7)( 4,22, 5)( 6,29,26, 8,18,36,30,15,17)( 9,12,31,40,11,14,19,21,28)(10,35,25)$ |
| 9A-1 | $9^{3},3^{4},1$ | $2880$ | $9$ | $32$ | $( 1,34,23)( 2,33,13,20,37,16,39,27,32)( 3, 7,24)( 4, 5,22)( 6,36,29,30,26,15, 8,17,18)( 9,14,12,19,31,21,40,28,11)(10,25,35)$ |
| 12A1 | $12^{2},4^{2},3,2^{2},1$ | $2160$ | $12$ | $32$ | $( 1,22, 4,10)( 2,39,20)( 3, 5,24,35)( 6,26,31,16, 8,36,14,13,30,17,28,32)( 7,34)( 9,18,21,37,40,15,12,33,19,29,11,27)(23,25)$ |
| 12A-1 | $12^{2},4^{2},3,2^{2},1$ | $2160$ | $12$ | $32$ | $( 1,22, 4,10)( 2,20,39)( 3, 5,24,35)( 6,36,28,16,30,26,14,32, 8,17,31,13)( 7,34)( 9,15,11,37,19,18,12,27,40,29,21,33)(23,25)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $25920=2^{6} \cdot 3^{4} \cdot 5$ | 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: | 25920.a | magma: IdentifyGroup(G);
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| Character table: |
| 1A | 2A | 2B | 3A1 | 3A-1 | 3B | 3C | 4A | 4B | 5A | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C | 6D | 9A1 | 9A-1 | 12A1 | 12A-1 | ||
| Size | 1 | 45 | 270 | 40 | 40 | 240 | 480 | 540 | 3240 | 5184 | 360 | 360 | 720 | 720 | 1440 | 2160 | 2880 | 2880 | 2160 | 2160 | |
| 2 P | 1A | 1A | 1A | 3A-1 | 3A1 | 3B | 3C | 2A | 2B | 5A | 3A1 | 3A-1 | 3B | 3B | 3C | 3B | 9A-1 | 9A1 | 6A1 | 6A-1 | |
| 3 P | 1A | 2A | 2B | 1A | 1A | 1A | 1A | 4A | 4B | 5A | 2A | 2A | 2A | 2A | 2A | 2B | 3A1 | 3A-1 | 4A | 4A | |
| 5 P | 1A | 2A | 2B | 3A-1 | 3A1 | 3B | 3C | 4A | 4B | 1A | 6A-1 | 6A1 | 6B-1 | 6B1 | 6C | 6D | 9A-1 | 9A1 | 12A-1 | 12A1 | |
| Type |
magma: CharacterTable(G);