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Properties

Label 40T24
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{10}:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(40, 24);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{10}:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $20$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,18)(2,17)(3,20)(4,19)(5,15)(6,16)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,40)(24,39)(25,36)(26,35)(27,33)(28,34)(29,32)(30,31), (1,13,2,14)(3,15,4,16)(5,11,6,12)(7,9,8,10)(17,37,18,38)(19,39,20,40)(21,36,22,35)(23,34,24,33)(25,31,26,32)(27,30,28,29), (1,14)(2,13)(3,15)(4,16)(5,11)(6,12)(7,10)(8,9)(17,37)(18,38)(19,40)(20,39)(21,36)(22,35)(23,33)(24,34)(25,31)(26,32)(27,29)(28,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T7 x 2, 20T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $D_{5}$

Degree 8: $D_4\times C_2$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T4, 20T11 x 2

Low degree siblings

40T24, 40T33 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 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, 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, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 4)( 5, 6)( 9,10)(13,14)(17,18)(21,22)(25,26)(29,30)(33,34)(39,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 3)( 2, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,35)(10,36)(11,34)(12,33)(13,32) (14,31)(15,30)(16,29)(17,27)(18,28)(19,26)(20,25)(21,24)(22,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5,37, 6,38)( 7,39, 8,40)( 9,36,10,35)(11,34,12,33)(13,31,14,32) (15,30,16,29)(17,28,18,27)(19,26,20,25)(21,23,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,37)(10,38)(11,39)(12,40)(13,35)(14,36)(15,34) (16,33)(17,32)(18,31)(19,29)(20,30)(21,28)(22,27)(23,26)(24,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,38,10,37)(11,39,12,40)(13,36,14,35)(15,34,16,33) (17,31,18,32)(19,29,20,30)(21,27,22,28)(23,26,24,25)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 7,11,15,20,24,28,31,36,38)( 2, 8,12,16,19,23,27,32,35,37)( 3, 5,10,14,18, 21,25,30,34,39)( 4, 6, 9,13,17,22,26,29,33,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 7,11,15,20,24,28,31,36,38)( 2, 8,12,16,19,23,27,32,35,37)( 3, 6,10,13,18, 22,25,29,34,40)( 4, 5, 9,14,17,21,26,30,33,39)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 8,11,16,20,23,28,32,36,37)( 2, 7,12,15,19,24,27,31,35,38)( 3, 5,10,14,18, 21,25,30,34,39)( 4, 6, 9,13,17,22,26,29,33,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 8,11,16,20,23,28,32,36,37)( 2, 7,12,15,19,24,27,31,35,38)( 3, 6,10,13,18, 22,25,29,34,40)( 4, 5, 9,14,17,21,26,30,33,39)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,11,20,28,36)( 2,12,19,27,35)( 3, 9,18,26,34, 4,10,17,25,33) ( 5,13,21,29,39, 6,14,22,30,40)( 7,15,24,31,38)( 8,16,23,32,37)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,20,28,36)( 2,12,19,27,35)( 3,10,18,25,34)( 4, 9,17,26,33) ( 5,14,21,30,39)( 6,13,22,29,40)( 7,15,24,31,38)( 8,16,23,32,37)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,12,20,27,36, 2,11,19,28,35)( 3, 9,18,26,34, 4,10,17,25,33)( 5,13,21,29,39, 6,14,22,30,40)( 7,16,24,32,38, 8,15,23,31,37)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,12,20,27,36, 2,11,19,28,35)( 3,10,18,25,34)( 4, 9,17,26,33) ( 5,14,21,30,39)( 6,13,22,29,40)( 7,16,24,32,38, 8,15,23,31,37)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,15,28,38,11,24,36, 7,20,31)( 2,16,27,37,12,23,35, 8,19,32)( 3,13,25,40,10, 22,34, 6,18,29)( 4,14,26,39, 9,21,33, 5,17,30)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,15,28,38,11,24,36, 7,20,31)( 2,16,27,37,12,23,35, 8,19,32)( 3,14,25,39,10, 21,34, 5,18,30)( 4,13,26,40, 9,22,33, 6,17,29)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,16,28,37,11,23,36, 8,20,32)( 2,15,27,38,12,24,35, 7,19,31)( 3,13,25,40,10, 22,34, 6,18,29)( 4,14,26,39, 9,21,33, 5,17,30)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,16,28,37,11,23,36, 8,20,32)( 2,15,27,38,12,24,35, 7,19,31)( 3,14,25,39,10, 21,34, 5,18,30)( 4,13,26,40, 9,22,33, 6,17,29)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,19,36,12,28, 2,20,35,11,27)( 3,17,34, 9,25, 4,18,33,10,26)( 5,22,39,13,30, 6,21,40,14,29)( 7,23,38,16,31, 8,24,37,15,32)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,19,36,12,28, 2,20,35,11,27)( 3,18,34,10,25)( 4,17,33, 9,26) ( 5,21,39,14,30)( 6,22,40,13,29)( 7,23,38,16,31, 8,24,37,15,32)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,20,36,11,28)( 2,19,35,12,27)( 3,17,34, 9,25, 4,18,33,10,26) ( 5,22,39,13,30, 6,21,40,14,29)( 7,24,38,15,31)( 8,23,37,16,32)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,20,36,11,28)( 2,19,35,12,27)( 3,18,34,10,25)( 4,17,33, 9,26) ( 5,21,39,14,30)( 6,22,40,13,29)( 7,24,38,15,31)( 8,23,37,16,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,25)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,32) (12,31)(13,33)(14,34)(15,35)(16,36)(17,40)(18,39)(19,38)(20,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,26)( 6,25)( 7,27)( 8,28)( 9,30)(10,29)(11,32) (12,31)(13,34)(14,33)(15,35)(16,36)(17,39)(18,40)(19,38)(20,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,24)( 2,23)( 3,21)( 4,22)( 5,25)( 6,26)( 7,28)( 8,27)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,36)(16,35)(17,40)(18,39)(19,37)(20,38)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $80=2^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  80.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A1 5A2 10A1 10A3 10B1 10B3 10C1 10C3 10D1 10D-1 10D3 10D-3 10E1 10E-1 10E3 10E-3
Size 1 1 1 1 2 2 10 10 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 5A2 5A1 5A2 5A1 5A2 5A1 5A2 5A1 5A2 5A1 5A2 5A1 5A1 5A2 5A1 5A2
5 P 1A 2B 2C 2A 2D 2E 2F 2G 4A 4B 1A 1A 2E 2E 2A 2A 2D 2D 2C 2D 2D 2C 2E 2B 2B 2E
Type
80.44.1a R 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 1
80.44.1b R 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 1
80.44.1c R 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 1
80.44.1d R 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 −1
80.44.1e R 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 −1
80.44.1f R 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 −1
80.44.1g R 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 −1
80.44.1h R 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 1
80.44.2a R 2 −2 −2 2 0 0 0 0 0 0 2 2 −2 −2 2 2 −2 −2 0 0 0 0 0 0 0 0
80.44.2b R 2 −2 2 −2 0 0 0 0 0 0 2 2 2 2 −2 −2 −2 −2 0 0 0 0 0 0 0 0
80.44.2c1 R 2 2 2 2 2 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
80.44.2c2 R 2 2 2 2 2 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
80.44.2d1 R 2 2 −2 −2 −2 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
80.44.2d2 R 2 2 −2 −2 −2 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
80.44.2e1 R 2 2 −2 −2 2 −2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
80.44.2e2 R 2 2 −2 −2 2 −2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
80.44.2f1 R 2 2 2 2 −2 −2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
80.44.2f2 R 2 2 2 2 −2 −2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
80.44.2g1 C 2 −2 −2 2 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52
80.44.2g2 C 2 −2 −2 2 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52
80.44.2g3 C 2 −2 −2 2 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52
80.44.2g4 C 2 −2 −2 2 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52
80.44.2h1 C 2 −2 2 −2 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52
80.44.2h2 C 2 −2 2 −2 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52
80.44.2h3 C 2 −2 2 −2 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52
80.44.2h4 C 2 −2 2 −2 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52

magma: CharacterTable(G);