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Magma
magma: G := TransitiveGroup(40, 3727);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $3727$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^5.C_2\wr C_5$ | ||
| 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));
| |
| Generators: | (1,15,5,39,12,2,16,6,40,11)(3,14,8,37,10,4,13,7,38,9)(17,30,23,33,28,18,29,24,34,27)(19,31,22,36,25,20,32,21,35,26), (1,29,40,27,33,21,10,19,7,13)(2,30,39,28,34,22,9,20,8,14)(3,31,37,26,36,23,12,17,5,15)(4,32,38,25,35,24,11,18,6,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5$: $C_5$ $10$: $C_{10}$ x 3 $20$: 20T3 $80$: $C_2^4 : C_5$ $160$: $C_2 \times (C_2^4 : C_5)$ x 3 $320$: 20T72 $2560$: 40T1853 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $C_5$
Degree 8: None
Degree 10: $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$ x 2
Degree 20: 20T40
Low degree siblings
40T3202 x 2, 40T3394 x 2, 40T3616 x 2, 40T3727, 40T3811 x 2, 40T3814 x 2Siblings 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 |
| $1^{40}$ | $1$ | $1$ | $()$ | |
| $2^{8},1^{24}$ | $5$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(21,22)(23,24)(29,30)(31,32)$ | |
| $2^{8},1^{24}$ | $5$ | $2$ | $( 1, 2)( 3, 4)(17,18)(19,20)(21,22)(23,24)(37,38)(39,40)$ | |
| $2^{16},1^{8}$ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(19,20)(21,22)(23,24)(25,26) (27,28)(29,30)(31,32)(37,38)(39,40)$ | |
| $4^{4},2^{8},1^{8}$ | $80$ | $4$ | $( 1, 3, 2, 4)( 7, 8)( 9,30,10,29)(11,31,12,32)(13,36)(14,35)(15,33)(16,34) (17,18)(21,23,22,24)(27,28)(39,40)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,24, 2,23)( 3,22, 4,21)( 5,27, 6,28)( 7,25, 8,26)( 9,30)(10,29)(11,32) (12,31)(13,33)(14,34)(15,35)(16,36)(37,38)(39,40)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,23, 2,24)( 3,21, 4,22)( 5,27, 6,28)( 7,25, 8,26)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,35)(16,36)(37,38)(39,40)$ | |
| $4^{4},2^{12}$ | $80$ | $4$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,30,10,29)(11,32,12,31)(13,16,14,15)(17,39) (18,40)(19,37)(20,38)(21,24)(22,23)(25,28)(26,27)(33,36,34,35)$ | |
| $5^{8}$ | $256$ | $5$ | $( 1, 5,12,16,40)( 2, 6,11,15,39)( 3, 8,10,13,38)( 4, 7, 9,14,37) (17,23,28,29,34)(18,24,27,30,33)(19,22,25,32,35)(20,21,26,31,36)$ | |
| $5^{8}$ | $256$ | $5$ | $( 1,12,40, 5,16)( 2,11,39, 6,15)( 3,10,38, 8,13)( 4, 9,37, 7,14) (17,28,34,23,29)(18,27,33,24,30)(19,25,35,22,32)(20,26,36,21,31)$ | |
| $5^{8}$ | $256$ | $5$ | $( 1,40,16,12, 5)( 2,39,15,11, 6)( 3,38,13,10, 8)( 4,37,14, 9, 7) (17,34,29,28,23)(18,33,30,27,24)(19,35,32,25,22)(20,36,31,26,21)$ | |
| $5^{8}$ | $256$ | $5$ | $( 1,16, 5,40,12)( 2,15, 6,39,11)( 3,13, 8,38,10)( 4,14, 7,37, 9) (17,29,23,34,28)(18,30,24,33,27)(19,32,22,35,25)(20,31,21,36,26)$ | |
| $2^{4},1^{32}$ | $5$ | $2$ | $(17,18)(19,20)(37,38)(39,40)$ | |
| $2^{12},1^{16}$ | $5$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(17,18)(19,20)(21,22)(23,24)(29,30)(31,32)(37,38) (39,40)$ | |
| $2^{12},1^{16}$ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(21,22)(23,24)(25,26)(27,28)(29,30) (31,32)$ | |
| $2^{20}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ | |
| $4^{4},2^{8},1^{8}$ | $80$ | $4$ | $( 1, 3, 2, 4)( 7, 8)( 9,30,10,29)(11,31,12,32)(13,36)(14,35)(15,33)(16,34) (19,20)(21,23,22,24)(27,28)(37,38)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,24, 2,23)( 3,22, 4,21)( 5,27, 6,28)( 7,25, 8,26)( 9,30)(10,29)(11,32) (12,31)(13,33)(14,34)(15,35)(16,36)(17,18)(19,20)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,23, 2,24)( 3,21, 4,22)( 5,27, 6,28)( 7,25, 8,26)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,35)(16,36)(17,18)(19,20)$ | |
| $4^{4},2^{12}$ | $80$ | $4$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,30,10,29)(11,32,12,31)(13,16,14,15)(17,40) (18,39)(19,38)(20,37)(21,24)(22,23)(25,28)(26,27)(33,36,34,35)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1, 5,12,16,40, 2, 6,11,15,39)( 3, 8,10,13,38, 4, 7, 9,14,37)(17,24,27,30,33, 18,23,28,29,34)(19,21,26,31,36,20,22,25,32,35)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,12,40, 6,15, 2,11,39, 5,16)( 3,10,38, 7,14, 4, 9,37, 8,13)(17,27,33,24,30, 18,28,34,23,29)(19,26,36,21,31,20,25,35,22,32)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,40,15,11, 6, 2,39,16,12, 5)( 3,38,14, 9, 7, 4,37,13,10, 8)(17,33,30,27,24, 18,34,29,28,23)(19,36,31,26,21,20,35,32,25,22)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,16, 5,40,11, 2,15, 6,39,12)( 3,13, 8,38, 9, 4,14, 7,37,10)(17,30,24,33,27, 18,29,23,34,28)(19,31,21,36,26,20,32,22,35,25)$ | |
| $4^{4},2^{10},1^{4}$ | $80$ | $4$ | $( 5,28)( 6,27)( 7,26)( 8,25)( 9,12,10,11)(13,36,14,35)(15,33,16,34)(17,39) (18,40)(19,38)(20,37)(21,22)(23,24)(29,31,30,32)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,22)( 2,21)( 3,23)( 4,24)( 7, 8)( 9,11)(10,12)(13,34,14,33)(15,36,16,35) (17,39,18,40)(19,37,20,38)(25,26)(29,31)(30,32)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,22)( 2,21)( 3,23)( 4,24)( 7, 8)( 9,12)(10,11)(13,34,14,33)(15,36,16,35) (17,40,18,39)(19,38,20,37)(25,26)(29,32)(30,31)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 3, 4)( 5,25, 6,26)( 7,28, 8,27)( 9,11)(10,12)(13,16,14,15)(17,18)(19,20) (23,24)(29,32)(30,31)(33,35,34,36)(37,38)(39,40)$ | |
| $4^{4},2^{6},1^{12}$ | $40$ | $4$ | $( 3, 4)( 5,25, 6,26)( 7,28, 8,27)( 9,12)(10,11)(13,16,14,15)(23,24)(29,31) (30,32)(33,35,34,36)$ | |
| $2^{20}$ | $16$ | $2$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,25)( 6,26)( 7,28)( 8,27)( 9,30)(10,29)(11,32) (12,31)(13,34)(14,33)(15,35)(16,36)(17,38)(18,37)(19,39)(20,40)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1, 5,29,36,37,21,25,10,15,18)( 2, 6,30,35,38,22,26, 9,16,17)( 3, 8,32,34,39, 23,27,11,14,19)( 4, 7,31,33,40,24,28,12,13,20)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,12,38,25,13,21,32,17, 6,33)( 2,11,37,26,14,22,31,18, 5,34)( 3,10,39,28,16, 23,30,19, 8,35)( 4, 9,40,27,15,24,29,20, 7,36)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,40,33,12, 8,22,20,14,32,28)( 2,39,34,11, 7,21,19,13,31,27)( 3,38,35,10, 6, 24,17,15,30,25)( 4,37,36, 9, 5,23,18,16,29,26)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,16,28,40,30,22,36, 7,20, 9)( 2,15,27,39,29,21,35, 8,19,10)( 3,13,26,37,31, 24,34, 6,18,12)( 4,14,25,38,32,23,33, 5,17,11)$ | |
| $4^{4},2^{10},1^{4}$ | $80$ | $4$ | $( 5,28)( 6,27)( 7,26)( 8,25)( 9,12,10,11)(13,36,14,35)(15,33,16,34)(17,40) (18,39)(19,37)(20,38)(21,22)(23,24)(29,31,30,32)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,22)( 2,21)( 3,23)( 4,24)( 7, 8)( 9,11)(10,12)(13,34,14,33)(15,36,16,35) (17,40,18,39)(19,38,20,37)(25,26)(29,31)(30,32)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 1,22)( 2,21)( 3,23)( 4,24)( 7, 8)( 9,12)(10,11)(13,34,14,33)(15,36,16,35) (17,39,18,40)(19,37,20,38)(25,26)(29,32)(30,31)$ | |
| $4^{4},2^{6},1^{12}$ | $40$ | $4$ | $( 3, 4)( 5,25, 6,26)( 7,28, 8,27)( 9,11)(10,12)(13,16,14,15)(23,24)(29,32) (30,31)(33,35,34,36)$ | |
| $4^{4},2^{10},1^{4}$ | $40$ | $4$ | $( 3, 4)( 5,25, 6,26)( 7,28, 8,27)( 9,12)(10,11)(13,16,14,15)(17,18)(19,20) (23,24)(29,31)(30,32)(33,35,34,36)(37,38)(39,40)$ | |
| $2^{20}$ | $16$ | $2$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,25)( 6,26)( 7,28)( 8,27)( 9,30)(10,29)(11,32) (12,31)(13,34)(14,33)(15,35)(16,36)(17,37)(18,38)(19,40)(20,39)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1, 5,29,36,37,22,26, 9,16,17)( 2, 6,30,35,38,21,25,10,15,18)( 3, 8,32,34,39, 24,28,12,13,20)( 4, 7,31,33,40,23,27,11,14,19)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,12,38,26,14,22,31,18, 6,33)( 2,11,37,25,13,21,32,17, 5,34)( 3,10,39,27,15, 24,29,20, 8,35)( 4, 9,40,28,16,23,30,19, 7,36)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,40,34,11, 7,21,19,14,32,28)( 2,39,33,12, 8,22,20,13,31,27)( 3,38,36, 9, 5, 23,18,15,30,25)( 4,37,35,10, 6,24,17,16,29,26)$ | |
| $10^{4}$ | $256$ | $10$ | $( 1,16,28,40,29,21,35, 8,19, 9)( 2,15,27,39,30,22,36, 7,20,10)( 3,13,26,37,32, 23,33, 5,17,12)( 4,14,25,38,31,24,34, 6,18,11)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5120=2^{10} \cdot 5$ | 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: | 5120.en | magma: IdentifyGroup(G);
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| Character table: | 44 x 44 character table |
magma: CharacterTable(G);