# Properties

 Label 40T14345 Order $$25920$$ n $$40$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSp(4,3)$

## Group action invariants

 Degree $n$ : $40$ Transitive number $t$ : $14345$ Group : $\PSp(4,3)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,4,7,10)(3,6,12,21,20)(5,9,16,27,38)(8,14,24,32,34)(11,19,23,31,17)(13,22,30,40,37)(15,25,28,18,29)(26,33,39,36,35), (1,3,4,7,12)(2,5,10,18,22)(6,9,16,27,38)(8,15,26,36,37)(11,20,30,40,34)(13,23,19,29,17)(14,25,28,21,31)(24,33,39,32,35) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Degree 2: None

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: None

Degree 20: None

## Low degree siblings

There are no siblings with degree $\leq 10$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 5, 5, 5, 5$ $5184$ $5$ $( 1,15,35,40,30)( 2,36, 7, 4, 6)( 3,37,34,20,33)( 5,39,23,18,12) ( 8,17,10,38,21)( 9,24,28,25,31)(11,22,19,27,16)(13,32,26,14,29)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1$ $40$ $3$ $( 1,31, 6)( 2,37,28)( 3,17,10)( 5,20, 8)( 7,22,29)( 9,33,27)(12,38,34) (13,19,18)(14,39,36)(15,24,40)(16,25,30)(26,32,35)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1$ $40$ $3$ $( 1, 6,31)( 2,28,37)( 3,10,17)( 5, 8,20)( 7,29,22)( 9,27,33)(12,34,38) (13,18,19)(14,36,39)(15,40,24)(16,30,25)(26,35,32)$ $9, 9, 9, 9, 3, 1$ $2880$ $9$ $( 1,40,26,31,15,32, 6,24,35)( 2,20,22,37, 8,29,28, 5, 7)( 3,36,38,17,14,34,10, 39,12)( 9,25,18,33,30,13,27,16,19)(11,21,23)$ $9, 9, 9, 9, 3, 1$ $2880$ $9$ $( 1,26,15, 6,35,40,31,32,24)( 2,22, 8,28, 7,20,37,29, 5)( 3,38,14,10,12,36,17, 34,39)( 9,18,30,27,19,25,33,13,16)(11,23,21)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $270$ $2$ $( 1,36)( 2,21)( 3,33)( 4,28)( 6,39)( 7,18)( 8,16)( 9,34)(10,29)(12,15)(13,27) (14,31)(17,24)(19,40)(20,26)(22,38)(23,37)(25,35)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $240$ $3$ $( 1,10,14)( 2, 8, 4)( 3,40,38)( 5,30,11)( 6,27,12)( 7,34,24)( 9,17,18) (13,15,39)(16,28,21)(19,22,33)(20,25,37)(23,26,35)(29,31,36)$ $6, 6, 6, 6, 6, 6, 3, 1$ $2160$ $6$ $( 1,31,10,36,14,29)( 2,28, 8,21, 4,16)( 3,22,40,33,38,19)( 5,11,30) ( 6,15,27,39,12,13)( 7,17,34,18,24, 9)(20,23,25,26,37,35)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $45$ $2$ $( 2,21)( 3,18)( 4,28)( 7,33)( 8,16)( 9,40)(17,38)(19,34)(20,26)(22,24)(23,37) (25,35)$ $6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1$ $360$ $6$ $( 1, 5,29)( 2,24, 9,21,22,40)( 3, 4,34,18,28,19)( 6,27,12)( 7,17,16,33,38, 8) (10,30,31)(11,36,14)(20,23,25,26,37,35)$ $6, 6, 6, 6, 3, 3, 3, 3, 1, 1, 1, 1$ $360$ $6$ $( 1,29, 5)( 2,40,22,21, 9,24)( 3,19,28,18,34, 4)( 6,12,27)( 7, 8,38,33,16,17) (10,31,30)(11,14,36)(20,35,37,26,25,23)$ $4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2$ $540$ $4$ $( 1, 6)( 2,33,21, 7)( 3,35,18,25)( 4,20,28,26)( 5,27)( 8,40,16, 9)(10,36) (11,31)(12,29)(13,32)(14,30)(15,39)(17,24,38,22)(19,37,34,23)$ $12, 12, 6, 6, 2, 2$ $2160$ $12$ $( 1,12, 5, 6,29,27)( 2,16,24,33, 9,38,21, 8,22, 7,40,17)( 3,37, 4,35,34,20,18, 23,28,25,19,26)(10,11,30,36,31,14)(13,32)(15,39)$ $12, 12, 6, 6, 2, 2$ $2160$ $12$ $( 1,27,29, 6, 5,12)( 2,38,40,33,22,16,21,17, 9, 7,24, 8)( 3,20,19,35,28,37,18, 26,34,25, 4,23)(10,14,31,36,30,11)(13,32)(15,39)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1$ $480$ $3$ $( 1,13,29)( 3, 7,25)( 4,24,38)( 6,39,30)( 9,19,37)(10,27,36)(11,15,12) (14,31,32)(17,28,22)(18,33,35)(23,40,34)$ $6, 6, 6, 3, 3, 3, 3, 3, 2, 2, 2, 1$ $1440$ $6$ $( 1,29,13)( 2,21)( 3,35, 7,18,25,33)( 4,17,24,28,38,22)( 6,30,39)( 8,16) ( 9,23,19,40,37,34)(10,36,27)(11,12,15)(14,32,31)(20,26)$ $6, 6, 6, 6, 3, 3, 3, 3, 3, 1$ $720$ $6$ $( 1,14,10)( 2,28, 8,21, 4,16)( 3,17,40,18,38, 9)( 5,11,30)( 6,12,27) ( 7,22,34,33,24,19)(13,39,15)(20,23,25,26,37,35)(29,36,31)$ $6, 6, 6, 6, 3, 3, 3, 3, 3, 1$ $720$ $6$ $( 1,10,14)( 2,16, 4,21, 8,28)( 3, 9,38,18,40,17)( 5,30,11)( 6,27,12) ( 7,19,24,33,34,22)(13,15,39)(20,35,37,26,25,23)(29,31,36)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1$ $3240$ $4$ $( 1, 6,12,29)( 2,33,35,18)( 3,21, 7,25)( 4,28)( 5,31,32,14)( 8,19,37, 9) (10,15,39,36)(11,27,30,13)(16,34,23,40)(17,24,38,22)$

## Group invariants

 Order: $25920=2^{6} \cdot 3^{4} \cdot 5$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 6 6 2 2 2 3 3 . . 4 3 3 2 2 5 2 3 . 1 1 3 4 2 3 2 2 4 4 2 2 1 2 2 1 1 1 1 . . 3 2 5 1 . . . . . . . . . . . . . . . . 1 . . 1a 2a 3a 6a 6b 3b 3c 9a 9b 4a 6c 6d 12a 12b 2b 6e 4b 5a 3d 6f 2P 1a 1a 3a 3a 3a 3c 3b 9b 9a 2a 3b 3c 6c 6d 1a 3a 2b 5a 3d 3d 3P 1a 2a 1a 2a 2a 1a 1a 3b 3c 4a 2a 2a 4a 4a 2b 2b 4b 5a 1a 2a 5P 1a 2a 3a 6b 6a 3c 3b 9b 9a 4a 6d 6c 12b 12a 2b 6e 4b 1a 3d 6f 7P 1a 2a 3a 6a 6b 3b 3c 9a 9b 4a 6c 6d 12a 12b 2b 6e 4b 5a 3d 6f 11P 1a 2a 3a 6b 6a 3c 3b 9b 9a 4a 6d 6c 12b 12a 2b 6e 4b 5a 3d 6f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 5 -3 -1 A -A C /C H /H 1 I /I -H -/H 1 1 -1 . 2 . X.3 5 -3 -1 -A A /C C /H H 1 /I I -/H -H 1 1 -1 . 2 . X.4 6 -2 3 1 1 -3 -3 . . 2 1 1 -1 -1 2 -1 . 1 . -2 X.5 10 2 1 -1 -1 D /D -/H -H 2 /C C H /H -2 1 . . 1 -1 X.6 10 2 1 -1 -1 /D D -H -/H 2 C /C /H H -2 1 . . 1 -1 X.7 15 7 . -2 -2 -3 -3 . . -1 1 1 -1 -1 3 . 1 . 3 1 X.8 15 -1 3 -1 -1 6 6 . . 3 2 2 . . -1 -1 -1 . . 2 X.9 20 4 5 1 1 2 2 -1 -1 . -2 -2 . . 4 1 . . -1 1 X.10 24 8 . 2 2 6 6 . . . 2 2 . . . . . -1 3 -1 X.11 30 -10 3 -1 -1 3 3 . . -2 -1 -1 1 1 2 -1 . . 3 -1 X.12 30 6 -3 -A A E /E . . 2 /I I /H H 2 -1 . . . . X.13 30 6 -3 A -A /E E . . 2 I /I H /H 2 -1 . . . . X.14 40 -8 -2 B /B F /F -H -/H . B /B . . . . . . 1 1 X.15 40 -8 -2 /B B /F F -/H -H . /B B . . . . . . 1 1 X.16 45 -3 . . . G /G . . 1 J /J -/H -H -3 . 1 . . . X.17 45 -3 . . . /G G . . 1 /J J -H -/H -3 . 1 . . . X.18 60 -4 -3 -1 -1 6 6 . . . 2 2 . . 4 1 . . -3 -1 X.19 64 . 4 . . -8 -8 1 1 . . . . . . . . -1 -2 . X.20 81 9 . . . . . . . -3 . . . . -3 . -1 1 . . A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 B = -2*E(3) = 1-Sqrt(-3) = 1-i3 C = E(3)-2*E(3)^2 = (1+3*Sqrt(-3))/2 = 2+3b3 D = 5*E(3)+2*E(3)^2 = (-7+3*Sqrt(-3))/2 = -2+3b3 E = 6*E(3)-3*E(3)^2 = (-3+9*Sqrt(-3))/2 = 3+9b3 F = 8*E(3)+2*E(3)^2 = -5+3*Sqrt(-3) = -5+3i3 G = -9*E(3)^2 = (9+9*Sqrt(-3))/2 = 9+9b3 H = -E(3) = (1-Sqrt(-3))/2 = -b3 I = 2*E(3)+E(3)^2 = (-3+Sqrt(-3))/2 = -1+b3 J = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3