# Properties

 Label 40T15 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times \OD_{16}$

Show commands: Magma

magma: G := TransitiveGroup(40, 15);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $15$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5\times \OD_{16}$ 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: (1,21,3,23,2,22,4,24)(5,26,7,28,6,25,8,27)(9,32,11,30,10,31,12,29)(13,35,16,34,14,36,15,33)(17,37,19,40,18,38,20,39), (1,14,25,37,10,24,33,6,18,30,3,15,27,40,12,21,35,8,20,31,2,13,26,38,9,23,34,5,17,29,4,16,28,39,11,22,36,7,19,32) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$5$:  $C_5$
$8$:  $C_4\times C_2$
$10$:  $C_{10}$ x 3
$16$:  $C_8:C_2$
$20$:  20T1 x 2, 20T3
$40$:  40T2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $C_5$

Degree 8: $C_8:C_2$

Degree 10: $C_{10}$

Degree 20: 20T1

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Label Cycle Type Size Order Index Representative 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 $2^{10},1^{20}$ $2$ $2$ $10$ $( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)(37,38)(39,40)$ 4A1 $4^{10}$ $1$ $4$ $30$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,32,30,31)(33,35,34,36)(37,40,38,39)$ 4A-1 $4^{10}$ $1$ $4$ $30$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,31,30,32)(33,36,34,35)(37,39,38,40)$ 4B $4^{10}$ $2$ $4$ $30$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,27,26,28)(29,31,30,32)(33,35,34,36)(37,39,38,40)$ 5A1 $5^{8}$ $1$ $5$ $32$ $( 1, 9,18,28,35)( 2,10,17,27,36)( 3,11,20,25,34)( 4,12,19,26,33)( 5,15,22,31,37)( 6,16,21,32,38)( 7,13,24,29,40)( 8,14,23,30,39)$ 5A-1 $5^{8}$ $1$ $5$ $32$ $( 1,18,35, 9,28)( 2,17,36,10,27)( 3,20,34,11,25)( 4,19,33,12,26)( 5,22,37,15,31)( 6,21,38,16,32)( 7,24,40,13,29)( 8,23,39,14,30)$ 5A2 $5^{8}$ $1$ $5$ $32$ $( 1,35,28,18, 9)( 2,36,27,17,10)( 3,34,25,20,11)( 4,33,26,19,12)( 5,37,31,22,15)( 6,38,32,21,16)( 7,40,29,24,13)( 8,39,30,23,14)$ 5A-2 $5^{8}$ $1$ $5$ $32$ $( 1,28, 9,35,18)( 2,27,10,36,17)( 3,25,11,34,20)( 4,26,12,33,19)( 5,31,15,37,22)( 6,32,16,38,21)( 7,29,13,40,24)( 8,30,14,39,23)$ 8A1 $8^{5}$ $2$ $8$ $35$ $( 1,21, 3,23, 2,22, 4,24)( 5,26, 7,28, 6,25, 8,27)( 9,32,11,30,10,31,12,29)(13,35,16,34,14,36,15,33)(17,37,19,40,18,38,20,39)$ 8A-1 $8^{5}$ $2$ $8$ $35$ $( 1,23, 4,21, 2,24, 3,22)( 5,28, 8,26, 6,27, 7,25)( 9,30,12,32,10,29,11,31)(13,34,15,35,14,33,16,36)(17,40,20,37,18,39,19,38)$ 8B1 $8^{5}$ $2$ $8$ $35$ $( 1,22, 4,23, 2,21, 3,24)( 5,26, 8,27, 6,25, 7,28)( 9,31,12,30,10,32,11,29)(13,35,15,33,14,36,16,34)(17,38,20,40,18,37,19,39)$ 8B-1 $8^{5}$ $2$ $8$ $35$ $( 1,24, 3,21, 2,23, 4,22)( 5,28, 7,25, 6,27, 8,26)( 9,29,11,32,10,30,12,31)(13,34,16,36,14,33,15,35)(17,39,19,37,18,40,20,38)$ 10A1 $10^{4}$ $1$ $10$ $36$ $( 1,27, 9,36,18, 2,28,10,35,17)( 3,26,11,33,20, 4,25,12,34,19)( 5,32,15,38,22, 6,31,16,37,21)( 7,30,13,39,24, 8,29,14,40,23)$ 10A-1 $10^{4}$ $1$ $10$ $36$ $( 1,17,35,10,28, 2,18,36, 9,27)( 3,19,34,12,25, 4,20,33,11,26)( 5,21,37,16,31, 6,22,38,15,32)( 7,23,40,14,29, 8,24,39,13,30)$ 10A3 $10^{4}$ $1$ $10$ $36$ $( 1,10,18,27,35, 2, 9,17,28,36)( 3,12,20,26,34, 4,11,19,25,33)( 5,16,22,32,37, 6,15,21,31,38)( 7,14,24,30,40, 8,13,23,29,39)$ 10A-3 $10^{4}$ $1$ $10$ $36$ $( 1,36,28,17, 9, 2,35,27,18,10)( 3,33,25,19,11, 4,34,26,20,12)( 5,38,31,21,15, 6,37,32,22,16)( 7,39,29,23,13, 8,40,30,24,14)$ 10B1 $10^{2},5^{4}$ $2$ $10$ $34$ $( 1, 9,18,28,35)( 2,10,17,27,36)( 3,11,20,25,34)( 4,12,19,26,33)( 5,16,22,32,37, 6,15,21,31,38)( 7,14,24,30,40, 8,13,23,29,39)$ 10B-1 $10^{2},5^{4}$ $2$ $10$ $34$ $( 1,28, 9,35,18)( 2,27,10,36,17)( 3,25,11,34,20)( 4,26,12,33,19)( 5,32,15,38,22, 6,31,16,37,21)( 7,30,13,39,24, 8,29,14,40,23)$ 10B3 $10^{2},5^{4}$ $2$ $10$ $34$ $( 1,35,28,18, 9)( 2,36,27,17,10)( 3,34,25,20,11)( 4,33,26,19,12)( 5,38,31,21,15, 6,37,32,22,16)( 7,39,29,23,13, 8,40,30,24,14)$ 10B-3 $10^{2},5^{4}$ $2$ $10$ $34$ $( 1,18,35, 9,28)( 2,17,36,10,27)( 3,20,34,11,25)( 4,19,33,12,26)( 5,21,37,16,31, 6,22,38,15,32)( 7,23,40,14,29, 8,24,39,13,30)$ 20A1 $20^{2}$ $1$ $20$ $38$ $( 1,33,27,20, 9, 4,36,25,18,12, 2,34,28,19,10, 3,35,26,17,11)( 5,39,32,24,15, 8,38,29,22,14, 6,40,31,23,16, 7,37,30,21,13)$ 20A-1 $20^{2}$ $1$ $20$ $38$ $( 1,26,10,34,18, 4,27,11,35,19, 2,25, 9,33,17, 3,28,12,36,20)( 5,30,16,40,22, 8,32,13,37,23, 6,29,15,39,21, 7,31,14,38,24)$ 20A3 $20^{2}$ $1$ $20$ $38$ $( 1,12,17,25,35, 4,10,20,28,33, 2,11,18,26,36, 3, 9,19,27,34)( 5,14,21,29,37, 8,16,24,31,39, 6,13,22,30,38, 7,15,23,32,40)$ 20A-3 $20^{2}$ $1$ $20$ $38$ $( 1,19,36,11,28, 4,17,34, 9,26, 2,20,35,12,27, 3,18,33,10,25)( 5,23,38,13,31, 8,21,40,15,30, 6,24,37,14,32, 7,22,39,16,29)$ 20A7 $20^{2}$ $1$ $20$ $38$ $( 1,11,17,26,35, 3,10,19,28,34, 2,12,18,25,36, 4, 9,20,27,33)( 5,13,21,30,37, 7,16,23,31,40, 6,14,22,29,38, 8,15,24,32,39)$ 20A-7 $20^{2}$ $1$ $20$ $38$ $( 1,34,27,19, 9, 3,36,26,18,11, 2,33,28,20,10, 4,35,25,17,12)( 5,40,32,23,15, 7,38,30,22,13, 6,39,31,24,16, 8,37,29,21,14)$ 20A9 $20^{2}$ $1$ $20$ $38$ $( 1,25,10,33,18, 3,27,12,35,20, 2,26, 9,34,17, 4,28,11,36,19)( 5,29,16,39,22, 7,32,14,37,24, 6,30,15,40,21, 8,31,13,38,23)$ 20A-9 $20^{2}$ $1$ $20$ $38$ $( 1,20,36,12,28, 3,17,33, 9,25, 2,19,35,11,27, 4,18,34,10,26)( 5,24,38,14,31, 7,21,39,15,29, 6,23,37,13,32, 8,22,40,16,30)$ 20B1 $20^{2}$ $2$ $20$ $38$ $( 1,11,17,26,35, 3,10,19,28,34, 2,12,18,25,36, 4, 9,20,27,33)( 5,14,21,29,37, 8,16,24,31,39, 6,13,22,30,38, 7,15,23,32,40)$ 20B-1 $20^{2}$ $2$ $20$ $38$ $( 1,34,27,19, 9, 3,36,26,18,11, 2,33,28,20,10, 4,35,25,17,12)( 5,39,32,24,15, 8,38,29,22,14, 6,40,31,23,16, 7,37,30,21,13)$ 20B3 $20^{2}$ $2$ $20$ $38$ $( 1,20,36,12,28, 3,17,33, 9,25, 2,19,35,11,27, 4,18,34,10,26)( 5,23,38,13,31, 8,21,40,15,30, 6,24,37,14,32, 7,22,39,16,29)$ 20B-3 $20^{2}$ $2$ $20$ $38$ $( 1,25,10,33,18, 3,27,12,35,20, 2,26, 9,34,17, 4,28,11,36,19)( 5,30,16,40,22, 8,32,13,37,23, 6,29,15,39,21, 7,31,14,38,24)$ 40A1 $40$ $2$ $40$ $39$ $( 1, 6,11,14,17,22,26,29,35,38, 3, 8,10,15,19,24,28,32,34,39, 2, 5,12,13,18,21,25,30,36,37, 4, 7, 9,16,20,23,27,31,33,40)$ 40A-1 $40$ $2$ $40$ $39$ $( 1,38,34,30,27,22,19,13, 9, 6, 3,39,36,31,26,24,18,16,11, 8, 2,37,33,29,28,21,20,14,10, 5, 4,40,35,32,25,23,17,15,12, 7)$ 40A3 $40$ $2$ $40$ $39$ $( 1,32,20, 8,36,22,12,40,28,16, 3,30,17, 5,33,24, 9,38,25,14, 2,31,19, 7,35,21,11,39,27,15, 4,29,18, 6,34,23,10,37,26,13)$ 40A-3 $40$ $2$ $40$ $39$ $( 1,30,19, 6,36,24,11,37,28,14, 4,32,17, 7,34,22, 9,39,26,16, 2,29,20, 5,35,23,12,38,27,13, 3,31,18, 8,33,21,10,40,25,15)$ 40A7 $40$ $2$ $40$ $39$ $( 1,14,26,38,10,24,34, 5,18,30, 4,16,27,40,11,22,35, 8,19,32, 2,13,25,37, 9,23,33, 6,17,29, 3,15,28,39,12,21,36, 7,20,31)$ 40A-7 $40$ $2$ $40$ $39$ $( 1,15,26,39,10,21,34, 7,18,31, 4,14,27,38,11,24,35, 5,19,30, 2,16,25,40, 9,22,33, 8,17,32, 3,13,28,37,12,23,36, 6,20,29)$ 40A9 $40$ $2$ $40$ $39$ $( 1,37,33,30,27,21,20,13, 9, 5, 4,39,36,32,25,24,18,15,12, 8, 2,38,34,29,28,22,19,14,10, 6, 3,40,35,31,26,23,17,16,11, 7)$ 40A-9 $40$ $2$ $40$ $39$ $( 1,40,34,32,27,23,19,15, 9, 7, 3,38,36,30,26,22,18,13,11, 6, 2,39,33,31,28,24,20,16,10, 8, 4,37,35,29,25,21,17,14,12, 5)$ 40B1 $40$ $2$ $40$ $39$ $( 1, 7,11,16,17,23,26,31,35,40, 3, 6,10,14,19,22,28,29,34,38, 2, 8,12,15,18,24,25,32,36,39, 4, 5, 9,13,20,21,27,30,33,37)$ 40B-1 $40$ $2$ $40$ $39$ $( 1, 8,12,16,17,24,25,31,35,39, 4, 6,10,13,20,22,28,30,33,38, 2, 7,11,15,18,23,26,32,36,40, 3, 5, 9,14,19,21,27,29,34,37)$ 40B3 $40$ $2$ $40$ $39$ $( 1,13,25,38,10,23,33, 5,18,29, 3,16,27,39,12,22,35, 7,20,32, 2,14,26,37, 9,24,34, 6,17,30, 4,15,28,40,11,21,36, 8,19,31)$ 40B-3 $40$ $2$ $40$ $39$ $( 1, 5,12,14,17,21,25,29,35,37, 4, 8,10,16,20,24,28,31,33,39, 2, 6,11,13,18,22,26,30,36,38, 3, 7, 9,15,19,23,27,32,34,40)$ 40B7 $40$ $2$ $40$ $39$ $( 1,16,25,39,10,22,33, 7,18,32, 3,14,27,37,12,24,35, 6,20,30, 2,15,26,40, 9,21,34, 8,17,31, 4,13,28,38,11,23,36, 5,19,29)$ 40B-7 $40$ $2$ $40$ $39$ $( 1,39,33,32,27,24,20,15, 9, 8, 4,38,36,29,25,22,18,14,12, 6, 2,40,34,31,28,23,19,16,10, 7, 3,37,35,30,26,21,17,13,11, 5)$ 40B9 $40$ $2$ $40$ $39$ $( 1,29,20, 6,36,23,12,37,28,13, 3,32,17, 8,33,22, 9,40,25,16, 2,30,19, 5,35,24,11,38,27,14, 4,31,18, 7,34,21,10,39,26,15)$ 40B-9 $40$ $2$ $40$ $39$ $( 1,31,19, 8,36,21,11,40,28,15, 4,30,17, 6,34,24, 9,37,26,14, 2,32,20, 7,35,22,12,39,27,16, 3,29,18, 5,33,23,10,38,25,13)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/10$

## 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: $2$ Label: 80.24 magma: IdentifyGroup(G); Character table: 50 x 50 character table

magma: CharacterTable(G);