# Properties

 Label 40T14344 Degree $40$ Order $25920$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSp(4,3)$

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Show commands: Magma

magma: G := TransitiveGroup(40, 14344);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $14344$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSp(4,3)$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,8,16,7)(2,6,13,19,17)(4,9,5,10,20)(11,23,36,26,35)(12,21,31,25,18)(14,27,29,22,34)(15,28,30,38,32)(24,37,40,39,33), (1,2,5,10,19)(3,4,7,15,28)(6,12,21,13,25)(8,16,29,38,14)(9,18,31,20,27)(11,22,33,37,34)(17,30,23,35,32)(24,26,39,40,36) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: None

Degree 20: None

## Low degree siblings

27T993, 36T12781, 40T14345, 45T666

Siblings are shown 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 $2^{16},1^{8}$ $45$ $2$ $16$ $( 2, 9)( 3,29)( 4,38)( 5,15)( 6,27)( 7,28)(10,31)(11,40)(14,35)(16,30)(17,22)(18,23)(20,21)(24,36)(26,34)(32,37)$ 2B $2^{20}$ $270$ $2$ $20$ $( 1,39)( 2,22)( 3,24)( 4,28)( 5,20)( 6,23)( 7,38)( 8,25)( 9,17)(10,14)(11,32)(12,19)(13,33)(15,21)(16,26)(18,27)(29,36)(30,34)(31,35)(37,40)$ 3A1 $3^{9},1^{13}$ $40$ $3$ $18$ $( 2,39,20)( 6, 8,30)( 9,40,19)(11,21,12)(13,32,16)(14,28,31)(15,29,18)(17,26,36)(27,37,33)$ 3A-1 $3^{9},1^{13}$ $40$ $3$ $18$ $( 2,20,39)( 6,30, 8)( 9,19,40)(11,12,21)(13,16,32)(14,31,28)(15,18,29)(17,36,26)(27,33,37)$ 3B $3^{12},1^{4}$ $240$ $3$ $24$ $( 1, 5,22)( 2,39,20)( 4,35,10)( 6, 8,30)( 9,13,15)(11,36,37)(12,26,27)(14,28,31)(16,18,19)(17,33,21)(23,25,34)(29,40,32)$ 3C $3^{12},1^{4}$ $480$ $3$ $24$ $( 1,25, 4)( 2,28,30)( 5,34,35)( 6,39,31)( 8,20,14)( 9,29,16)(10,22,23)(11,17,27)(12,36,33)(13,40,18)(15,32,19)(21,26,37)$ 4A $4^{8},2^{2},1^{4}$ $540$ $4$ $26$ $( 2, 5, 9,15)( 3,14,29,35)( 4,37,38,32)( 6,30,27,16)( 7,11,28,40)( 8,25)(10,36,31,24)(13,33)(17,21,22,20)(18,26,23,34)$ 4B $4^{10}$ $3240$ $4$ $30$ $( 1,13,39,33)( 2,20,22, 5)( 3,28,24, 4)( 6,34,23,30)( 7,36,38,29)( 8,12,25,19)( 9,21,17,15)(10,11,14,32)(16,27,26,18)(31,40,35,37)$ 5A $5^{8}$ $5184$ $5$ $32$ $( 1,33,14,26,37)( 2,13,23, 6,39)( 3,20, 5,28,17)( 4,24,32,30,35)( 7,34,29,27,18)( 8,16,19,15,21)( 9,31,38,36,11)(10,22,40,25,12)$ 6A1 $6^{4},3,2^{4},1^{5}$ $360$ $6$ $26$ $( 1, 4)( 2,20,39)( 3,24)( 5,35)( 6,31, 8,14,30,28)( 9,21,40,12,19,11)(10,22)(13,17,32,26,16,36)(15,33,29,27,18,37)$ 6A-1 $6^{4},3,2^{4},1^{5}$ $360$ $6$ $26$ $( 1, 4)( 2,39,20)( 3,24)( 5,35)( 6,28,30,14, 8,31)( 9,11,19,12,40,21)(10,22)(13,36,16,26,32,17)(15,37,18,27,29,33)$ 6B1 $6^{5},3^{2},2,1^{2}$ $720$ $6$ $30$ $( 1, 6, 5, 8,22,30)( 2,34,39,23,20,25)( 4,14,35,28,10,31)( 7,38)( 9,16,13,18,15,19)(11,37,36)(12,17,26,33,27,21)(29,32,40)$ 6B-1 $6^{5},3^{2},2,1^{2}$ $720$ $6$ $30$ $( 1,30,22, 8, 5, 6)( 2,25,20,23,39,34)( 4,31,10,28,35,14)( 7,38)( 9,19,15,18,13,16)(11,36,37)(12,21,27,33,26,17)(29,40,32)$ 6C $6^{5},3^{2},2,1^{2}$ $1440$ $6$ $30$ $( 1,36,25,33, 4,12)( 2,30,28)( 5,27,34,11,35,17)( 6,20,39,14,31, 8)( 9,16,29)(10,37,22,21,23,26)(13,32,40,19,18,15)(24,38)$ 6D $6^{6},2^{2}$ $2160$ $6$ $32$ $( 1, 2, 5,39,22,20)( 3,24)( 4,14,35,28,10,31)( 6,34, 8,23,30,25)( 7,38)( 9,21,13,17,15,33)(11,40,36,32,37,29)(12,18,26,19,27,16)$ 9A1 $9^{3},3^{4},1$ $2880$ $9$ $32$ $( 1,23,34)( 2,13,37,39,32,33,20,16,27)( 3,24, 7)( 4,22, 5)( 6,29,26, 8,18,36,30,15,17)( 9,12,31,40,11,14,19,21,28)(10,35,25)$ 9A-1 $9^{3},3^{4},1$ $2880$ $9$ $32$ $( 1,34,23)( 2,33,13,20,37,16,39,27,32)( 3, 7,24)( 4, 5,22)( 6,36,29,30,26,15, 8,17,18)( 9,14,12,19,31,21,40,28,11)(10,25,35)$ 12A1 $12^{2},4^{2},3,2^{2},1$ $2160$ $12$ $32$ $( 1,22, 4,10)( 2,39,20)( 3, 5,24,35)( 6,26,31,16, 8,36,14,13,30,17,28,32)( 7,34)( 9,18,21,37,40,15,12,33,19,29,11,27)(23,25)$ 12A-1 $12^{2},4^{2},3,2^{2},1$ $2160$ $12$ $32$ $( 1,22, 4,10)( 2,20,39)( 3, 5,24,35)( 6,36,28,16,30,26,14,32, 8,17,31,13)( 7,34)( 9,15,11,37,19,18,12,27,40,29,21,33)(23,25)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/16$

## Group invariants

 Order: $25920=2^{6} \cdot 3^{4} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 25920.a magma: IdentifyGroup(G); Character table:

 1A 2A 2B 3A1 3A-1 3B 3C 4A 4B 5A 6A1 6A-1 6B1 6B-1 6C 6D 9A1 9A-1 12A1 12A-1 Size 1 45 270 40 40 240 480 540 3240 5184 360 360 720 720 1440 2160 2880 2880 2160 2160 2 P 1A 1A 1A 3A-1 3A1 3B 3C 2A 2B 5A 3A1 3A-1 3B 3B 3C 3B 9A-1 9A1 6A1 6A-1 3 P 1A 2A 2B 1A 1A 1A 1A 4A 4B 5A 2A 2A 2A 2A 2A 2B 3A1 3A-1 4A 4A 5 P 1A 2A 2B 3A-1 3A1 3B 3C 4A 4B 1A 6A-1 6A1 6B-1 6B1 6C 6D 9A-1 9A1 12A-1 12A1 Type

magma: CharacterTable(G);