LMFDB - Galois group: 40T42

Show commands: Magma

magma: G := TransitiveGroup(40, 42);

Group action invariants

Degree $n$:	$40$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$42$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_5:C_8$
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:	(1,24,3,21,2,23,4,22)(5,33,40,11,6,34,39,12)(7,36,37,9,8,35,38,10)(13,20,32,28,14,19,31,27)(15,17,29,25,16,18,30,26), (1,18,35,9,26,4,19,33,12,27,2,17,36,10,25,3,20,34,11,28)(5,22,38,14,32,7,23,40,15,30,6,21,37,13,31,8,24,39,16,29)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$8$: $C_8$ x 2, $C_4\times C_2$
$16$: $C_8\times C_2$
$20$: $F_5$
$40$: $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$C_8$ x 2, $C_4\times C_2$
$16$:	$C_8\times C_2$
$20$:	$F_5$
$40$:	$F_{5}\times C_2$

Subfields

Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $F_5$
Degree 8: $C_8$
Degree 10: $F_5$
Degree 20: 20T9

Low degree siblings

40T42
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:	80.28	magma: IdentifyGroup(G);

Character table:

      2  4  4  4  4  4  4  4  4   4   4   4   4   4   4   4   4   2   2   2  2
      5  1  .  1  .  1  .  1  .   .   .   .   .   .   .   .   .   1   1   1  1

        1a 2a 2b 2c 4a 4b 4c 4d  8a  8b  8c  8d  8e  8f  8g  8h 20a 20b 10a 5a
     2P 1a 1a 1a 1a 2b 2b 2b 2b  4b  4b  4b  4b  4d  4d  4d  4d 10a 10a  5a 5a
     3P 1a 2a 2b 2c 4c 4d 4a 4b  8h  8g  8f  8e  8d  8c  8b  8a 20b 20a 10a 5a
     5P 1a 2a 2b 2c 4a 4b 4c 4d  8c  8d  8a  8b  8g  8h  8e  8f  4a  4c  2b 1a
     7P 1a 2a 2b 2c 4c 4d 4a 4b  8f  8e  8h  8g  8b  8a  8d  8c 20b 20a 10a 5a
    11P 1a 2a 2b 2c 4c 4d 4a 4b  8h  8g  8f  8e  8d  8c  8b  8a 20b 20a 10a 5a
    13P 1a 2a 2b 2c 4a 4b 4c 4d  8c  8d  8a  8b  8g  8h  8e  8f 20a 20b 10a 5a
    17P 1a 2a 2b 2c 4a 4b 4c 4d  8a  8b  8c  8d  8e  8f  8g  8h 20a 20b 10a 5a
    19P 1a 2a 2b 2c 4c 4d 4a 4b  8h  8g  8f  8e  8d  8c  8b  8a 20b 20a 10a 5a

X.1      1  1  1  1  1  1  1  1   1   1   1   1   1   1   1   1   1   1   1  1
X.2      1 -1  1 -1 -1  1 -1  1  -1   1  -1   1   1  -1   1  -1  -1  -1   1  1
X.3      1 -1  1 -1 -1  1 -1  1   1  -1   1  -1  -1   1  -1   1  -1  -1   1  1
X.4      1  1  1  1  1  1  1  1  -1  -1  -1  -1  -1  -1  -1  -1   1   1   1  1
X.5      1 -1 -1  1  A -A -A  A   C  -C  -C   C -/C  /C  /C -/C   A  -A  -1  1
X.6      1 -1 -1  1  A -A -A  A  -C   C   C  -C  /C -/C -/C  /C   A  -A  -1  1
X.7      1 -1 -1  1 -A  A  A -A -/C  /C  /C -/C   C  -C  -C   C  -A   A  -1  1
X.8      1 -1 -1  1 -A  A  A -A  /C -/C -/C  /C  -C   C   C  -C  -A   A  -1  1
X.9      1 -1  1 -1  1 -1  1 -1   A  -A   A  -A   A  -A   A  -A   1   1   1  1
X.10     1 -1  1 -1  1 -1  1 -1  -A   A  -A   A  -A   A  -A   A   1   1   1  1
X.11     1  1 -1 -1  A  A -A -A -/C -/C  /C  /C  -C  -C   C   C   A  -A  -1  1
X.12     1  1 -1 -1  A  A -A -A  /C  /C -/C -/C   C   C  -C  -C   A  -A  -1  1
X.13     1  1 -1 -1 -A -A  A  A   C   C  -C  -C  /C  /C -/C -/C  -A   A  -1  1
X.14     1  1 -1 -1 -A -A  A  A  -C  -C   C   C -/C -/C  /C  /C  -A   A  -1  1
X.15     1  1  1  1 -1 -1 -1 -1   A   A   A   A  -A  -A  -A  -A  -1  -1   1  1
X.16     1  1  1  1 -1 -1 -1 -1  -A  -A  -A  -A   A   A   A   A  -1  -1   1  1
X.17     4  .  4  . -4  . -4  .   .   .   .   .   .   .   .   .   1   1  -1 -1
X.18     4  .  4  .  4  .  4  .   .   .   .   .   .   .   .   .  -1  -1  -1 -1
X.19     4  . -4  .  B  . -B  .   .   .   .   .   .   .   .   .   A  -A   1 -1
X.20     4  . -4  . -B  .  B  .   .   .   .   .   .   .   .   .  -A   A   1 -1

A = -E(4)
  = -Sqrt(-1) = -i
B = 4*E(4)
  = 4*Sqrt(-1) = 4i
C = -E(8)

magma: CharacterTable(G);

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Group invariants

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	40T42
Degree	$40$
Order	$80$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_5:C_8$