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Properties

Label 40T1479
Degree $40$
Order $1600$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5^2:C_2^2$

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Copy content magma:G := TransitiveGroup(40, 1479);
 

Group invariants

Abstract group:  $F_5^2:C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $1600=2^{6} \cdot 5^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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Nilpotency class:   not nilpotent
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Group action invariants

Degree $n$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1479$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,23,9,40,17,15,26,32,33,7)(2,24,10,39,18,16,25,31,34,8)(3,21,11,38,19,13,28,30,36,5)(4,22,12,37,20,14,27,29,35,6)$, $(1,23,12,14,34,31,28,38)(2,24,11,13,33,32,27,37)(3,21,9,15,35,29,25,39)(4,22,10,16,36,30,26,40)(5,17,7,20,6,18,8,19)$, $(1,15,27,37,18,31,36,5)(2,16,28,38,17,32,35,6)(3,13,26,40,20,29,34,8)(4,14,25,39,19,30,33,7)(9,23,12,22,10,24,11,21)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:  $C_4\wr C_2$ x 2, $C_2 \times (C_2^2:C_4)$
$64$:  $C_4^2:C_2^2$
$800$:  $F_5 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: None

Degree 8: $C_4\wr C_2$

Degree 10: $F_5 \wr C_2$

Degree 20: 20T155

Low degree siblings

20T212 x 2, 20T215 x 2, 40T1479, 40T1485 x 2, 40T1509, 40T1512, 40T1517, 40T1520, 40T1554 x 2, 40T1558 x 2, 40T1563 x 2, 40T1566 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{40}$ $1$ $1$ $0$ $()$
2A $2^{20}$ $1$ $2$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
2B $2^{10},1^{20}$ $10$ $2$ $10$ $( 5,22)( 6,21)( 7,24)( 8,23)(13,14)(15,16)(29,38)(30,37)(31,40)(32,39)$
2C $2^{18},1^{4}$ $10$ $2$ $18$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(25,34)(26,33)(27,35)(28,36)(29,30)(31,32)(37,38)(39,40)$
2D $2^{20}$ $20$ $2$ $20$ $( 1,39)( 2,40)( 3,37)( 4,38)( 5,20)( 6,19)( 7,18)( 8,17)( 9,24)(10,23)(11,22)(12,21)(13,35)(14,36)(15,34)(16,33)(25,32)(26,31)(27,30)(28,29)$
2E $2^{20}$ $20$ $2$ $20$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,34)( 6,33)( 7,36)( 8,35)( 9,22)(10,21)(11,23)(12,24)(17,29)(18,30)(19,32)(20,31)(25,38)(26,37)(27,39)(28,40)$
2F $2^{16},1^{8}$ $25$ $2$ $16$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,38)( 6,37)( 7,40)( 8,39)(13,30)(14,29)(15,32)(16,31)(17,33)(18,34)(19,36)(20,35)$
2G $2^{20}$ $25$ $2$ $20$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,37)( 6,38)( 7,39)( 8,40)(13,29)(14,30)(15,31)(16,32)(17,34)(18,33)(19,35)(20,36)(21,22)(23,24)(25,26)(27,28)$
4A1 $4^{9},2^{2}$ $10$ $4$ $29$ $( 1,34,17,25)( 2,33,18,26)( 3,35,19,27)( 4,36,20,28)( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)(21,23,22,24)(29,31,30,32)(37,39,38,40)$
4A-1 $4^{9},2^{2}$ $10$ $4$ $29$ $( 1,25,17,34)( 2,26,18,33)( 3,27,19,35)( 4,28,20,36)( 5, 8, 6, 7)( 9,10)(11,12)(13,16,14,15)(21,24,22,23)(29,32,30,31)(37,40,38,39)$
4B1 $4^{9},1^{4}$ $10$ $4$ $27$ $( 1, 3, 2, 4)( 5,21,13,38)( 6,22,14,37)( 7,23,15,40)( 8,24,16,39)( 9,11,10,12)(17,19,18,20)(25,27,26,28)(33,36,34,35)$
4B-1 $4^{9},1^{4}$ $10$ $4$ $27$ $( 1, 4, 2, 3)( 5,38,13,21)( 6,37,14,22)( 7,40,15,23)( 8,39,16,24)( 9,12,10,11)(17,20,18,19)(25,28,26,27)(33,35,34,36)$
4C1 $4^{10}$ $25$ $4$ $30$ $( 1,20,10,36)( 2,19, 9,35)( 3,17,12,34)( 4,18,11,33)( 5,32,37,16)( 6,31,38,15)( 7,29,39,13)( 8,30,40,14)(21,23,22,24)(25,28,26,27)$
4C-1 $4^{10}$ $25$ $4$ $30$ $( 1,36,10,20)( 2,35, 9,19)( 3,34,12,17)( 4,33,11,18)( 5,16,37,32)( 6,15,38,31)( 7,13,39,29)( 8,14,40,30)(21,24,22,23)(25,27,26,28)$
4D1 $4^{10}$ $25$ $4$ $30$ $( 1, 4, 2, 3)( 5,32,22,39)( 6,31,21,40)( 7,29,24,38)( 8,30,23,37)( 9,27,34,19)(10,28,33,20)(11,26,35,18)(12,25,36,17)(13,15,14,16)$
4D-1 $4^{10}$ $25$ $4$ $30$ $( 1, 3, 2, 4)( 5,39,22,32)( 6,40,21,31)( 7,38,24,29)( 8,37,23,30)( 9,19,34,27)(10,20,33,28)(11,18,35,26)(12,17,36,25)(13,16,14,15)$
4E $4^{10}$ $50$ $4$ $30$ $( 1,28,18,35)( 2,27,17,36)( 3,25,20,33)( 4,26,19,34)( 5, 7, 6, 8)( 9,11,10,12)(13,23,37,31)(14,24,38,32)(15,22,39,30)(16,21,40,29)$
4F $4^{10}$ $50$ $4$ $30$ $( 1, 4, 2, 3)( 5,31,37,15)( 6,32,38,16)( 7,30,39,14)( 8,29,40,13)( 9,27,34,19)(10,28,33,20)(11,26,35,18)(12,25,36,17)(21,24,22,23)$
4G1 $4^{9},1^{4}$ $50$ $4$ $27$ $( 1,33,17,26)( 2,34,18,25)( 3,36,19,28)( 4,35,20,27)( 5,32, 6,31)( 7,29, 8,30)(13,23,14,24)(15,22,16,21)(37,39,38,40)$
4G-1 $4^{9},1^{4}$ $50$ $4$ $27$ $( 1,26,17,33)( 2,25,18,34)( 3,28,19,36)( 4,27,20,35)( 5,31, 6,32)( 7,30, 8,29)(13,24,14,23)(15,21,16,22)(37,40,38,39)$
4H1 $4^{9},2^{2}$ $50$ $4$ $29$ $( 1,36, 2,35)( 3,34, 4,33)( 5,37,21,29)( 6,38,22,30)( 7,39,23,31)( 8,40,24,32)( 9,28,10,27)(11,25,12,26)(13,14)(15,16)(17,19,18,20)$
4H-1 $4^{9},2^{2}$ $50$ $4$ $29$ $( 1,35, 2,36)( 3,33, 4,34)( 5,29,21,37)( 6,30,22,38)( 7,31,23,39)( 8,32,24,40)( 9,27,10,28)(11,26,12,25)(13,14)(15,16)(17,20,18,19)$
4I $4^{10}$ $100$ $4$ $30$ $( 1,15, 2,16)( 3,13, 4,14)( 5,20,22,28)( 6,19,21,27)( 7,18,24,26)( 8,17,23,25)( 9,40,34,31)(10,39,33,32)(11,38,35,29)(12,37,36,30)$
4J $4^{10}$ $100$ $4$ $30$ $( 1,16,18,23)( 2,15,17,24)( 3,14,20,21)( 4,13,19,22)( 5,36,29,27)( 6,35,30,28)( 7,33,31,25)( 8,34,32,26)( 9,39,10,40)(11,37,12,38)$
5A $5^{4},1^{20}$ $8$ $5$ $16$ $( 5,21,38,13,30)( 6,22,37,14,29)( 7,23,40,15,32)( 8,24,39,16,31)$
5B $5^{8}$ $16$ $5$ $32$ $( 1,26, 9,33,17)( 2,25,10,34,18)( 3,28,11,36,19)( 4,27,12,35,20)( 5,38,30,21,13)( 6,37,29,22,14)( 7,40,32,23,15)( 8,39,31,24,16)$
8A1 $8^{5}$ $100$ $8$ $35$ $( 1,16, 4,13, 2,15, 3,14)( 5,25,32,36,22,17,39,12)( 6,26,31,35,21,18,40,11)( 7,28,29,33,24,20,38,10)( 8,27,30,34,23,19,37, 9)$
8A-1 $8^{5}$ $100$ $8$ $35$ $( 1,14, 3,15, 2,13, 4,16)( 5,12,39,17,22,36,32,25)( 6,11,40,18,21,35,31,26)( 7,10,38,20,24,33,29,28)( 8, 9,37,19,23,34,30,27)$
8B1 $8^{5}$ $100$ $8$ $35$ $( 1,23,19,38,10,31,35,14)( 2,24,20,37, 9,32,36,13)( 3,21,18,39,12,29,33,15)( 4,22,17,40,11,30,34,16)( 5,25, 8,27, 6,26, 7,28)$
8B-1 $8^{5}$ $100$ $8$ $35$ $( 1,14,35,31,10,38,19,23)( 2,13,36,32, 9,37,20,24)( 3,15,33,29,12,39,18,21)( 4,16,34,30,11,40,17,22)( 5,28, 7,26, 6,27, 8,25)$
10A $10^{2},2^{10}$ $8$ $10$ $28$ $( 1, 2)( 3, 4)( 5,14,21,29,38, 6,13,22,30,37)( 7,16,23,31,40, 8,15,24,32,39)( 9,10)(11,12)(17,18)(19,20)(25,26)(27,28)(33,34)(35,36)$
10B $10^{4}$ $16$ $10$ $36$ $( 1,34,26,18, 9, 2,33,25,17,10)( 3,35,28,20,11, 4,36,27,19,12)( 5,22,38,14,30, 6,21,37,13,29)( 7,24,40,16,32, 8,23,39,15,31)$
10C $5^{4},2^{10}$ $40$ $10$ $26$ $( 1,33,26,17, 9)( 2,34,25,18,10)( 3,36,28,19,11)( 4,35,27,20,12)( 5,22)( 6,21)( 7,24)( 8,23)(13,14)(15,16)(29,38)(30,37)(31,40)(32,39)$
10D $10^{2},2^{8},1^{4}$ $40$ $10$ $26$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,14,21,29,38, 6,13,22,30,37)( 7,16,23,31,40, 8,15,24,32,39)(25,34)(26,33)(27,35)(28,36)$
10E $10^{4}$ $80$ $10$ $36$ $( 1,31, 9,16,17,39,26,24,33, 8)( 2,32,10,15,18,40,25,23,34, 7)( 3,29,11,14,19,37,28,22,36, 6)( 4,30,12,13,20,38,27,21,35, 5)$
10F $10^{4}$ $80$ $10$ $36$ $( 1,29,33,22,26,14,17, 6, 9,37)( 2,30,34,21,25,13,18, 5,10,38)( 3,32,36,23,28,15,19, 7,11,40)( 4,31,35,24,27,16,20, 8,12,39)$
20A1 $20,4^{4},2^{2}$ $40$ $20$ $33$ $( 1,34,17,25)( 2,33,18,26)( 3,35,19,27)( 4,36,20,28)( 5,32,14,39,21, 7,29,16,38,23, 6,31,13,40,22, 8,30,15,37,24)( 9,10)(11,12)$
20A-1 $20,4^{4},2^{2}$ $40$ $20$ $33$ $( 1,18,26,10)( 2,17,25, 9)( 3,20,28,12)( 4,19,27,11)( 5,31,14,40,21, 8,29,15,38,24, 6,32,13,39,22, 7,30,16,37,23)(33,34)(35,36)$
20B1 $20,4^{4},1^{4}$ $40$ $20$ $31$ $( 1,36,25,20, 9, 3,34,27,17,11, 2,35,26,19,10, 4,33,28,18,12)( 5,21,13,38)( 6,22,14,37)( 7,23,15,40)( 8,24,16,39)$
20B-1 $20,4^{4},1^{4}$ $40$ $20$ $31$ $( 1,35,25,19, 9, 4,34,28,17,12, 2,36,26,20,10, 3,33,27,18,11)( 5,21,30,13)( 6,22,29,14)( 7,23,32,15)( 8,24,31,16)$

Malle's constant $a(G)$:     $1/10$

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Character table

40 x 40 character table

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Regular extensions

Data not computed