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Properties

Label 40T44
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(40, 44);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2\times F_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,17,35,10,25,4,20,34,12,28)(2,18,36,9,26,3,19,33,11,27)(5,22,37,13,31,8,24,40,16,30)(6,21,38,14,32,7,23,39,15,29), (1,21,4,23)(2,22,3,24)(5,36,40,9)(6,35,39,10)(7,34,38,12)(8,33,37,11)(13,18,31,26)(14,17,32,25)(15,20,29,28)(16,19,30,27), (1,5,17,13)(2,6,18,14)(3,7,19,15)(4,8,20,16)(9,29,11,32)(10,30,12,31)(21,26,38,33)(22,25,37,34)(23,27,39,36)(24,28,40,35)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 5: $F_5$

Degree 8: $C_4\times C_2$

Degree 10: $F_5$, $F_{5}\times C_2$ x 2

Degree 20: 20T9 x 2, 20T13

Low degree siblings

20T16 x 4, 40T44 x 2, 40T56

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

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:  80.50
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 5A 10A 10B 10C
Size 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2D 2D 2D 2D 2D 2D 2D 2D 5A 5A 5A 5A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A-1 4A1 4D-1 4C1 4B-1 4D1 4C-1 4B1 1A 2A 2B 2C
Type
80.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.50.1b R 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1
80.50.1c R 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1
80.50.1d R 1 −1 1 −1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 1 −1 1 −1
80.50.1e R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 −1 1 −1
80.50.1f R 1 1 −1 −1 1 −1 1 −1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1
80.50.1g R 1 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
80.50.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
80.50.1i1 C 1 −1 −1 1 −1 1 1 −1 −i i i −i −i i −i i 1 −1 −1 1
80.50.1i2 C 1 −1 −1 1 −1 1 1 −1 i −i −i i i −i i −i 1 −1 −1 1
80.50.1j1 C 1 −1 1 −1 −1 −1 1 1 −i i i −i i −i i −i 1 −1 1 −1
80.50.1j2 C 1 −1 1 −1 −1 −1 1 1 i −i −i i −i i −i i 1 −1 1 −1
80.50.1k1 C 1 1 −1 −1 −1 1 −1 1 −i i −i i i −i −i i 1 1 −1 −1
80.50.1k2 C 1 1 −1 −1 −1 1 −1 1 i −i i −i −i i i −i 1 1 −1 −1
80.50.1l1 C 1 1 1 1 −1 −1 −1 −1 −i i −i i −i i i −i 1 1 1 1
80.50.1l2 C 1 1 1 1 −1 −1 −1 −1 i −i i −i i −i −i i 1 1 1 1
80.50.4a R 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1
80.50.4b R 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 1 −1
80.50.4c R 4 −4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 −1 1
80.50.4d R 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1

magma: CharacterTable(G);