↑

Properties

Label 40T1450
Degree $40$
Order $1600$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{10}^2.C_2^2$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(40, 1450);
 

Group invariants

Abstract group:  $D_{10}^2.C_2^2$
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
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1450$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,28,36,9)(2,27,35,10)(3,26,33,12)(4,25,34,11)(5,24,32,15)(6,23,31,16)(7,21,29,14)(8,22,30,13)(17,20,18,19)(37,40,38,39)$, $(1,15,17,8,35,38,11,29,26,23)(2,16,18,7,36,37,12,30,25,24)(3,13,19,6,34,40,9,32,27,21)(4,14,20,5,33,39,10,31,28,22)$, $(1,20,11,33)(2,19,12,34)(3,18,9,36)(4,17,10,35)(5,7)(6,8)(13,23,40,29)(14,24,39,30)(15,21,38,32)(16,22,37,31)(25,27)(26,28)$, $(1,13,26,21,11,32,35,40,17,6)(2,14,25,22,12,31,36,39,18,5)(3,15,27,23,9,29,34,38,19,8)(4,16,28,24,10,30,33,37,20,7)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $Q_8:C_2$ x 2, $C_2^4$
$32$:  $Q_8:C_2^2$ x 2, $C_2 \times (C_4\times C_2):C_2$
$64$:  16T82
$400$:  $(D_5 \wr C_2):C_2$
$800$:  20T154

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 8: $Q_8:C_2^2$

Degree 10: $(D_5 \wr C_2):C_2$

Degree 20: 20T154

Low degree siblings

40T1270 x 2, 40T1312 x 2, 40T1313 x 2, 40T1318 x 4, 40T1321 x 2, 40T1450 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $2^{20}$ $2$ $2$ $20$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$
2C $2^{20}$ $20$ $2$ $20$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,25)( 6,26)( 7,28)( 8,27)( 9,38)(10,37)(11,40)(12,39)(17,21)(18,22)(19,23)(20,24)(29,34)(30,33)(31,36)(32,35)$
2D $2^{20}$ $20$ $2$ $20$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,10)( 6, 9)( 7,12)( 8,11)(13,27)(14,28)(15,26)(16,25)(17,29)(18,30)(19,32)(20,31)(33,39)(34,40)(35,38)(36,37)$
2E $2^{20}$ $20$ $2$ $20$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,27)( 6,28)( 7,25)( 8,26)( 9,22)(10,21)(11,23)(12,24)(13,20)(14,19)(15,17)(16,18)(33,40)(34,39)(35,38)(36,37)$
2F $2^{14},1^{12}$ $20$ $2$ $14$ $( 3, 4)( 5,14)( 6,13)( 7,15)( 8,16)( 9,10)(19,20)(21,40)(22,39)(23,37)(24,38)(27,28)(29,30)(33,34)$
2G $2^{20}$ $25$ $2$ $20$ $( 1,36)( 2,35)( 3,33)( 4,34)( 5,32)( 6,31)( 7,29)( 8,30)( 9,28)(10,27)(11,25)(12,26)(13,22)(14,21)(15,24)(16,23)(17,18)(19,20)(37,38)(39,40)$
2H $2^{16},1^{8}$ $25$ $2$ $16$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,22)( 6,21)( 7,24)( 8,23)(17,35)(18,36)(19,34)(20,33)(29,38)(30,37)(31,39)(32,40)$
2I $2^{20}$ $50$ $2$ $20$ $( 1,34)( 2,33)( 3,35)( 4,36)( 5,16)( 6,15)( 7,14)( 8,13)( 9,26)(10,25)(11,27)(12,28)(17,19)(18,20)(21,38)(22,37)(23,40)(24,39)(29,32)(30,31)$
4A $4^{10}$ $20$ $4$ $30$ $( 1,33, 2,34)( 3,35, 4,36)( 5, 8, 6, 7)( 9,26,10,25)(11,28,12,27)(13,16,14,15)(17,20,18,19)(21,24,22,23)(29,32,30,31)(37,39,38,40)$
4B $4^{10}$ $20$ $4$ $30$ $( 1,21, 2,22)( 3,24, 4,23)( 5,26, 6,25)( 7,28, 8,27)( 9,30,10,29)(11,32,12,31)(13,36,14,35)(15,34,16,33)(17,40,18,39)(19,37,20,38)$
4C1 $4^{8},2^{4}$ $50$ $4$ $28$ $( 1,34,11,19)( 2,33,12,20)( 3,35, 9,17)( 4,36,10,18)( 5,29,22,38)( 6,30,21,37)( 7,32,24,40)( 8,31,23,39)(13,16)(14,15)(25,28)(26,27)$
4C-1 $4^{8},2^{4}$ $50$ $4$ $28$ $( 1,19,11,34)( 2,20,12,33)( 3,17, 9,35)( 4,18,10,36)( 5,38,22,29)( 6,37,21,30)( 7,40,24,32)( 8,39,23,31)(13,16)(14,15)(25,28)(26,27)$
4D1 $4^{8},2^{2},1^{4}$ $50$ $4$ $26$ $( 1,26,35,11)( 2,25,36,12)( 3,27,34, 9)( 4,28,33,10)( 5, 6)( 7, 8)(13,22,40,31)(14,21,39,32)(15,24,38,30)(16,23,37,29)$
4D-1 $4^{8},2^{2},1^{4}$ $50$ $4$ $26$ $( 1,11,35,26)( 2,12,36,25)( 3, 9,34,27)( 4,10,33,28)( 5, 6)( 7, 8)(13,31,40,22)(14,32,39,21)(15,30,38,24)(16,29,37,23)$
4E $4^{10}$ $100$ $4$ $30$ $( 1,15,36,24)( 2,16,35,23)( 3,13,33,22)( 4,14,34,21)( 5,27,32,10)( 6,28,31, 9)( 7,26,29,12)( 8,25,30,11)(17,38,18,37)(19,40,20,39)$
4F $4^{10}$ $100$ $4$ $30$ $( 1,39,18,21)( 2,40,17,22)( 3,37,20,23)( 4,38,19,24)( 5,25,13,35)( 6,26,14,36)( 7,28,15,34)( 8,27,16,33)( 9,30,10,29)(11,31,12,32)$
4G $4^{10}$ $100$ $4$ $30$ $( 1,24,18,29)( 2,23,17,30)( 3,21,20,31)( 4,22,19,32)( 5, 9, 6,10)( 7,12, 8,11)(13,33,39,27)(14,34,40,28)(15,35,37,25)(16,36,38,26)$
4H $4^{8},2^{2},1^{4}$ $100$ $4$ $26$ $( 1,12,35,25)( 2,11,36,26)( 3, 9,34,27)( 4,10,33,28)( 7, 8)(13,21,40,32)(14,22,39,31)(15,24,38,30)(16,23,37,29)(17,18)$
4I $4^{8},2^{4}$ $100$ $4$ $28$ $( 1,21,35,40)( 2,22,36,39)( 3,24,34,37)( 4,23,33,38)( 5,25,14,12)( 6,26,13,11)( 7,27,16, 9)( 8,28,15,10)(17,32)(18,31)(19,30)(20,29)$
4J $4^{10}$ $100$ $4$ $30$ $( 1,10,36,27)( 2, 9,35,28)( 3,11,33,25)( 4,12,34,26)( 5,30,40,15)( 6,29,39,16)( 7,32,38,14)( 8,31,37,13)(17,20,18,19)(21,23,22,24)$
5A $5^{8}$ $8$ $5$ $32$ $( 1,11,17,26,35)( 2,12,18,25,36)( 3, 9,19,27,34)( 4,10,20,28,33)( 5,22,39,14,31)( 6,21,40,13,32)( 7,24,37,16,30)( 8,23,38,15,29)$
5B $5^{8}$ $8$ $5$ $32$ $( 1,35,26,17,11)( 2,36,25,18,12)( 3,34,27,19, 9)( 4,33,28,20,10)( 5,14,22,31,39)( 6,13,21,32,40)( 7,16,24,30,37)( 8,15,23,29,38)$
5C $5^{4},1^{20}$ $8$ $5$ $16$ $( 1,17,35,11,26)( 2,18,36,12,25)( 3,19,34, 9,27)( 4,20,33,10,28)$
10A $10^{4}$ $8$ $10$ $36$ $( 1,27,11,34,17, 3,26, 9,35,19)( 2,28,12,33,18, 4,25,10,36,20)( 5,16,22,30,39, 7,14,24,31,37)( 6,15,21,29,40, 8,13,23,32,38)$
10B $10^{4}$ $8$ $10$ $36$ $( 1,12,17,25,35, 2,11,18,26,36)( 3,10,19,28,34, 4, 9,20,27,33)( 5,21,39,13,31, 6,22,40,14,32)( 7,23,37,15,30, 8,24,38,16,29)$
10C $10^{4}$ $8$ $10$ $36$ $( 1,28,11,33,17, 4,26,10,35,20)( 2,27,12,34,18, 3,25, 9,36,19)( 5,15,22,29,39, 8,14,23,31,38)( 6,16,21,30,40, 7,13,24,32,37)$
10D $10^{2},2^{10}$ $8$ $10$ $28$ $( 1, 2)( 3, 4)( 5,32,14,40,22, 6,31,13,39,21)( 7,29,16,38,24, 8,30,15,37,23)( 9,10)(11,12)(17,18)(19,20)(25,26)(27,28)(33,34)(35,36)$
10E $10^{4}$ $8$ $10$ $36$ $( 1,36,26,18,11, 2,35,25,17,12)( 3,33,27,20, 9, 4,34,28,19,10)( 5,40,31,21,14, 6,39,32,22,13)( 7,38,30,23,16, 8,37,29,24,15)$
10F $10^{4}$ $16$ $10$ $36$ $( 1,19,35, 9,26, 3,17,34,11,27)( 2,20,36,10,25, 4,18,33,12,28)( 5,30,14,37,22, 7,31,16,39,24)( 6,29,13,38,21, 8,32,15,40,23)$
10G $10^{2},2^{10}$ $16$ $10$ $28$ $( 1, 9,17,27,35, 3,11,19,26,34)( 2,10,18,28,36, 4,12,20,25,33)( 5, 7)( 6, 8)(13,15)(14,16)(21,23)(22,24)(29,32)(30,31)(37,39)(38,40)$
10H1 $10^{4}$ $40$ $10$ $36$ $( 1,21,35,40,26,13,17,32,11, 6)( 2,22,36,39,25,14,18,31,12, 5)( 3,23,34,38,27,15,19,29, 9, 8)( 4,24,33,37,28,16,20,30,10, 7)$
10H3 $10^{4}$ $40$ $10$ $36$ $( 1,40,17, 6,35,13,11,21,26,32)( 2,39,18, 5,36,14,12,22,25,31)( 3,38,19, 8,34,15, 9,23,27,29)( 4,37,20, 7,33,16,10,24,28,30)$
10I1 $10^{4}$ $40$ $10$ $36$ $( 1,29,35, 8,26,23,17,38,11,15)( 2,30,36, 7,25,24,18,37,12,16)( 3,32,34, 6,27,21,19,40, 9,13)( 4,31,33, 5,28,22,20,39,10,14)$
10I3 $10^{4}$ $40$ $10$ $36$ $( 1, 8,17,15,35,23,11,29,26,38)( 2, 7,18,16,36,24,12,30,25,37)( 3, 6,19,13,34,21, 9,32,27,40)( 4, 5,20,14,33,22,10,31,28,39)$
10J $10^{4}$ $80$ $10$ $36$ $( 1, 8,11,38,17,29,26,23,35,15)( 2, 7,12,37,18,30,25,24,36,16)( 3, 5, 9,39,19,31,27,22,34,14)( 4, 6,10,40,20,32,28,21,33,13)$
10K $10,5^{2},2^{9},1^{2}$ $80$ $10$ $26$ $( 1,17,35,11,26)( 2,18,36,12,25)( 3,20,34,10,27, 4,19,33, 9,28)( 5,14)( 6,13)( 7,15)( 8,16)(21,40)(22,39)(23,37)(24,38)(29,30)$
20A $20,4^{5}$ $80$ $20$ $34$ $( 1,33, 2,34)( 3,35, 4,36)( 5,38,32,24,14, 8,40,30,22,15, 6,37,31,23,13, 7,39,29,21,16)( 9,26,10,25)(11,28,12,27)(17,20,18,19)$
20B $20^{2}$ $80$ $20$ $38$ $( 1,40,36,31,26,21,18,14,11, 6, 2,39,35,32,25,22,17,13,12, 5)( 3,37,33,29,27,24,20,15, 9, 7, 4,38,34,30,28,23,19,16,10, 8)$

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(G);
 

Character table

40 x 40 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed