# Properties

 Label 45T666 Degree $45$ Order $25920$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSp(4,3)$

Show commands: Magma

magma: G := TransitiveGroup(45, 666);

## Group action invariants

 Degree $n$: $45$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $666$ 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: (1,3,6,12,19)(2,5,10,18,11)(4,8,15,22,29)(7,14,20,28,40)(9,16,24,33,13)(17,25,34,36,31)(21,30,41,45,32)(23,35,42,37,39)(26,27,38,44,43), (1,2,4,7,13)(3,5,9,14,8)(6,11,15,21,29)(10,17,24,31,19)(12,18,26,37,43)(16,23,34,35,33)(20,27,38,44,40)(22,32,41,45,30)(25,36,42,28,39) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 3: None

Degree 5: None

Degree 9: None

Degree 15: None

## Low degree siblings

27T993, 36T12781, 40T14344, 40T14345

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 $1^{45}$ $1$ $1$ $0$ $()$ 2A $2^{16},1^{13}$ $45$ $2$ $16$ $( 1,10)( 3,15)( 4, 5)( 7,11)( 8,26)(12,20)(13,18)(14,17)(16,19)(22,41)(23,24)(27,37)(29,32)(33,42)(34,36)(39,43)$ 2B $2^{20},1^{5}$ $270$ $2$ $20$ $( 1,34)( 2,38)( 3, 5)( 4,15)( 6,44)( 7,29)( 8,14)(10,36)(11,32)(12,41)(13,24)(16,33)(17,26)(18,23)(19,42)(20,22)(21,31)(25,30)(27,39)(37,43)$ 3A1 $3^{12},1^{9}$ $40$ $3$ $24$ $( 2,12,39)( 3,27,13)( 4, 6,34)( 5,22,31)( 7,19,45)( 9,16,29)(10,24,26)(11,28,32)(14,25,43)(17,20,38)(21,41,36)(33,35,42)$ 3A-1 $3^{12},1^{9}$ $40$ $3$ $24$ $( 2,39,12)( 3,13,27)( 4,34, 6)( 5,31,22)( 7,45,19)( 9,29,16)(10,26,24)(11,32,28)(14,43,25)(17,38,20)(21,36,41)(33,42,35)$ 3B $3^{13},1^{6}$ $240$ $3$ $26$ $( 1,15,44)( 2,10, 5)( 3,38,36)( 4, 6,34)( 7,16,42)( 8,30,37)( 9,35,45)(12,24,22)(13,20,41)(14,25,43)(17,21,27)(19,29,33)(26,31,39)$ 3C $3^{14},1^{3}$ $480$ $3$ $28$ $( 1,15,44)( 2, 9,36)( 3,10,35)( 5,45,38)( 7,17,22)( 8,37,30)(11,28,32)(12,16,21)(13,26,33)(14,43,25)(18,40,23)(19,20,31)(24,42,27)(29,41,39)$ 4A $4^{8},2^{6},1$ $540$ $4$ $30$ $( 1, 8,10,26)( 2,25)( 3,43,15,39)( 4,27, 5,37)( 6,31)( 7,22,11,41)( 9,45)(12,29,20,32)(13,16,18,19)(14,36,17,34)(21,44)(23,42,24,33)(28,40)(30,38)$ 4B $4^{10},2^{2},1$ $3240$ $4$ $32$ $( 1,26,34,17)( 2,31,38,21)( 3,37, 5,43)( 4,39,15,27)( 6,25,44,30)( 7,18,29,23)( 8,36,14,10)( 9,45)(11,13,32,24)(12,19,41,42)(16,22,33,20)(28,40)$ 5A $5^{9}$ $5184$ $5$ $36$ $( 1,10,14,38,26)( 2,37, 3,15,39)( 4,45, 5,24,32)( 6,20,30,25,19)( 7,36, 9,34,18)( 8,29,31,13,43)(11,21,23,17,27)(12,16,33,35,41)(22,40,44,28,42)$ 6A1 $6^{4},3^{4},2^{4},1$ $360$ $6$ $32$ $( 1,30)( 2,39,12)( 3,35,27,42,13,33)( 4,14, 6,25,34,43)( 5,31,22)( 7,20,19,38,45,17)( 8,44)( 9,21,16,41,29,36)(10,26,24)(11,32,28)(15,37)(23,40)$ 6A-1 $6^{4},3^{4},2^{4},1$ $360$ $6$ $32$ $( 1,30)( 2,12,39)( 3,33,13,42,27,35)( 4,43,34,25, 6,14)( 5,22,31)( 7,17,45,38,19,20)( 8,44)( 9,36,29,41,16,21)(10,24,26)(11,28,32)(15,37)(23,40)$ 6B1 $6^{5},3^{3},2,1^{4}$ $720$ $6$ $32$ $( 1,25,15,43,44,14)( 2,26,10,31, 5,39)( 3,27,38,17,36,21)( 4,37, 6, 8,34,30)( 7,19,16,29,42,33)( 9,45,35)(11,32)(12,22,24)(13,41,20)$ 6B-1 $6^{5},3^{3},2,1^{4}$ $720$ $6$ $32$ $( 1,14,44,43,15,25)( 2,39, 5,31,10,26)( 3,21,36,17,38,27)( 4,30,34, 8, 6,37)( 7,33,42,29,16,19)( 9,35,45)(11,32)(12,24,22)(13,20,41)$ 6C $6^{5},3^{4},2,1$ $1440$ $6$ $34$ $( 1,44,15)( 2, 5, 9,45,36,38)( 3,35,10)( 4,34)( 7,42,17,27,22,24)( 8,28,37,32,30,11)(12,21,16)(13,41,26,39,33,29)(14,40,43,23,25,18)(19,31,20)$ 6D $6^{6},3,2^{2},1^{2}$ $2160$ $6$ $34$ $( 1, 6,15,34,44, 4)( 2, 3,10,38, 5,36)( 7,19,16,29,42,33)( 8,43,30,14,37,25)( 9,45,35)(11,32)(12,20,24,41,22,13)(17,39,21,26,27,31)(18,23)$ 9A1 $9^{4},3^{3}$ $2880$ $9$ $38$ $( 1,18, 8)( 2,31,27,12, 5,13,39,22, 3)( 4,32,25, 6,11,43,34,28,14)( 7,26,16,19,10,29,45,24, 9)(15,40,37)(17,42,41,20,33,36,38,35,21)(23,30,44)$ 9A-1 $9^{4},3^{3}$ $2880$ $9$ $38$ $( 1, 8,18)( 2,13,31,39,27,22,12, 3, 5)( 4,43,32,34,25,28, 6,14,11)( 7,29,26,45,16,24,19, 9,10)(15,37,40)(17,36,42,38,41,35,20,21,33)(23,44,30)$ 12A1 $12^{2},6^{2},4^{2},1$ $2160$ $12$ $38$ $( 1,37,30,15)( 2,24,39,10,12,26)( 3,41,35,29,27,36,42, 9,13,21,33,16)( 4, 7,14,20, 6,19,25,38,34,45,43,17)( 5,28,31,11,22,32)( 8,23,44,40)$ 12A-1 $12^{2},6^{2},4^{2},1$ $2160$ $12$ $38$ $( 1,37,30,15)( 2,26,12,10,39,24)( 3,36,33,29,13,41,42,16,27,21,35, 9)( 4,19,43,20,34, 7,25,17, 6,45,14,38)( 5,32,22,11,31,28)( 8,23,44,40)$

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);