# Properties

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

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

## 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 5 2 2 3 3 3 . . 6 4 3 3 2 2 . 1 1 2 2 3 4 1 3 1 . 4 4 2 2 2 1 2 2 1 1 . 3 2 2 2 5 1 . . . . . . . . . . . . . . 1 . . . . 1a 2a 3a 6a 4a 3b 3c 9a 9b 2b 4b 6b 6c 12a 12b 5a 3d 6d 6e 6f 2P 1a 1a 3a 3a 2a 3c 3b 9b 9a 1a 2b 3c 3b 6b 6c 5a 3d 3d 3a 3a 3P 1a 2a 1a 2a 4a 1a 1a 3b 3c 2b 4b 2b 2b 4b 4b 5a 1a 2b 2b 2b 5P 1a 2a 3a 6a 4a 3c 3b 9b 9a 2b 4b 6c 6b 12b 12a 1a 3d 6d 6f 6e 7P 1a 2a 3a 6a 4a 3b 3c 9a 9b 2b 4b 6b 6c 12a 12b 5a 3d 6d 6e 6f 11P 1a 2a 3a 6a 4a 3c 3b 9b 9a 2b 4b 6c 6b 12b 12a 5a 3d 6d 6f 6e 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 1 -1 1 -1 A /A F /F -3 1 G /G -/F -F . 2 . J -J X.3 5 1 -1 1 -1 /A A /F F -3 1 /G G -F -/F . 2 . -J J X.4 6 2 3 -1 . -3 -3 . . -2 2 1 1 -1 -1 1 . -2 1 1 X.5 10 -2 1 1 . B /B -F -/F 2 2 /A A F /F . 1 -1 -1 -1 X.6 10 -2 1 1 . /B B -/F -F 2 2 A /A /F F . 1 -1 -1 -1 X.7 15 3 . . 1 -3 -3 . . 7 -1 1 1 -1 -1 . 3 1 -2 -2 X.8 15 -1 3 -1 -1 6 6 . . -1 3 2 2 . . . . 2 -1 -1 X.9 20 4 5 1 . 2 2 -1 -1 4 . -2 -2 . . . -1 1 1 1 X.10 24 . . . . 6 6 . . 8 . 2 2 . . -1 3 -1 2 2 X.11 30 2 3 -1 . 3 3 . . -10 -2 -1 -1 1 1 . 3 -1 -1 -1 X.12 30 2 -3 -1 . C /C . . 6 2 G /G /F F . . . J -J X.13 30 2 -3 -1 . /C C . . 6 2 /G G F /F . . . -J J X.14 40 . -2 . . D /D -F -/F -8 . H /H . . . 1 1 /H H X.15 40 . -2 . . /D D -/F -F -8 . /H H . . . 1 1 H /H X.16 45 -3 . . 1 E /E . . -3 1 I /I -/F -F . . . . . X.17 45 -3 . . 1 /E E . . -3 1 /I I -F -/F . . . . . X.18 60 4 -3 1 . 6 6 . . -4 . 2 2 . . . -3 -1 -1 -1 X.19 64 . 4 . . -8 -8 1 1 . . . . . . -1 -2 . . . X.20 81 -3 . . -1 . . . . 9 -3 . . . . 1 . . . . A = -2*E(3)+E(3)^2 = (1-3*Sqrt(-3))/2 = -1-3b3 B = 5*E(3)+2*E(3)^2 = (-7+3*Sqrt(-3))/2 = -2+3b3 C = 6*E(3)-3*E(3)^2 = (-3+9*Sqrt(-3))/2 = 3+9b3 D = 2*E(3)+8*E(3)^2 = -5-3*Sqrt(-3) = -5-3i3 E = -9*E(3)^2 = (9+9*Sqrt(-3))/2 = 9+9b3 F = -E(3)^2 = (1+Sqrt(-3))/2 = 1+b3 G = 2*E(3)+E(3)^2 = (-3+Sqrt(-3))/2 = -1+b3 H = -2*E(3) = 1-Sqrt(-3) = 1-i3 I = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 J = E(3)-E(3)^2 = Sqrt(-3) = i3