# Properties

 Label 45T666 Order $$25920$$ n $$45$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSp(4,3)$

## Group action invariants

 Degree $n$ : $45$ Transitive number $t$ : $666$ Group : $\PSp(4,3)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) 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) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Degree 3: None

Degree 5: None

Degree 9: None

Degree 15: None

## Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 5, 5, 5, 5, 5$ $5184$ $5$ $( 1,14,41,38,43)( 2, 8,19,24,42)( 3,37,34,30,13)( 4, 9,26,15,17) ( 5,25,29,12, 7)( 6,21,44,35,31)(10,27,33,28,32)(11,22,40,18,16) (20,23,36,45,39)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $45$ $2$ $( 3,45)( 5,35)( 6,34)( 8,28)(10,38)(11,30)(12,27)(14,40)(16,24)(18,25)(19,39) (20,29)(21,42)(23,43)(31,41)(32,37)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $40$ $3$ $( 1, 9,36)( 2,17,33)( 5,23,19)( 6, 8,32)( 7,15,22)(10,27,40)(12,14,38) (13,26,44)(18,21,29)(20,25,42)(28,37,34)(35,43,39)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $40$ $3$ $( 1,36, 9)( 2,33,17)( 5,19,23)( 6,32, 8)( 7,22,15)(10,40,27)(12,38,14) (13,44,26)(18,29,21)(20,42,25)(28,34,37)(35,39,43)$ $4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1$ $540$ $4$ $( 1,13)( 2, 7)( 3,31,45,41)( 5,32,35,37)( 6,43,34,23)( 8,39,28,19)( 9,26) (10,42,38,21)(11,16,30,24)(12,29,27,20)(14,18,40,25)(15,17)(22,33)(36,44)$ $6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1$ $360$ $6$ $( 1,36, 9)( 2,33,17)( 3,45)( 5,39,23,35,19,43)( 6,37, 8,34,32,28)( 7,22,15) (10,14,27,38,40,12)(11,30)(13,44,26)(16,24)(18,20,21,25,29,42)(31,41)$ $6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1$ $360$ $6$ $( 1, 9,36)( 2,17,33)( 3,45)( 5,43,19,35,23,39)( 6,28,32,34, 8,37)( 7,15,22) (10,12,40,38,27,14)(11,30)(13,26,44)(16,24)(18,42,29,25,21,20)(31,41)$ $12, 12, 6, 6, 4, 4, 1$ $2160$ $12$ $( 1,26,36,13, 9,44)( 2,15,33, 7,17,22)( 3,41,45,31)( 5,34,39,32,23,28,35, 6, 19,37,43, 8)(10,29,14,42,27,18,38,20,40,21,12,25)(11,24,30,16)$ $12, 12, 6, 6, 4, 4, 1$ $2160$ $12$ $( 1,44, 9,13,36,26)( 2,22,17, 7,33,15)( 3,41,45,31)( 5,28,43,32,19,34,35, 8, 23,37,39, 6)(10,18,12,42,40,29,38,25,27,21,14,20)(11,24,30,16)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1$ $240$ $3$ $( 1,13, 2)( 3,41,11)( 5,19,23)( 6,20,10)( 8,25,27)( 9,26,17)(12,28,18) (14,37,21)(29,38,34)(30,45,31)(32,42,40)(33,36,44)(35,39,43)$ $6, 6, 6, 6, 6, 3, 3, 3, 2, 1, 1, 1, 1$ $720$ $6$ $( 1,32,13,42, 2,40)( 3,31,41,30,11,45)( 4,16)( 5,35,19,39,23,43) ( 6,26,20,17,10, 9)( 8,44,25,33,27,36)(12,18,28)(14,21,37)(29,34,38)$ $6, 6, 6, 6, 6, 3, 3, 3, 2, 1, 1, 1, 1$ $720$ $6$ $( 1,40, 2,42,13,32)( 3,45,11,30,41,31)( 4,16)( 5,43,23,39,19,35) ( 6, 9,10,17,20,26)( 8,36,27,33,25,44)(12,28,18)(14,37,21)(29,38,34)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $480$ $3$ $( 1,44,17)( 2,36,26)( 3,41,11)( 5,23,19)( 6,42,27)( 7,22,15)( 8,20,40) ( 9,13,33)(10,32,25)(12,34,21)(14,28,29)(18,38,37)(30,45,31)(35,43,39)$ $6, 6, 6, 6, 6, 3, 3, 3, 3, 2, 1$ $1440$ $6$ $( 1, 6,44,42,17,27)( 2,10,36,32,26,25)( 3,31,41,30,11,45)( 4,16) ( 5,43,23,39,19,35)( 7,15,22)( 8,13,20,33,40, 9)(12,21,34)(14,29,28)(18,37,38)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $270$ $2$ $( 1,27)( 2,25)( 3,23)( 4,16)( 5,41)( 6,17)( 7,22)( 8,13)( 9,20)(10,26)(11,19) (12,21)(14,28)(18,37)(30,35)(31,43)(32,36)(33,40)(39,45)(42,44)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1$ $3240$ $4$ $( 1,41,27, 5)( 2,19,23, 8)( 3,13,25,11)( 4,37,17,14)( 6,28,16,18)( 7,26,21, 9) (10,12,20,22)(15,24)(29,38)(30,33,36,43)(31,35,40,32)(39,44,45,42)$ $6, 6, 6, 6, 6, 6, 3, 2, 2, 1, 1$ $2160$ $6$ $( 1,20, 2, 6,13,10)( 3,23,11,19,41, 5)( 4,16)( 7,15)( 8,36,27,33,25,44) ( 9,32,17,40,26,42)(12,28,18)(14,38,21,29,37,34)(30,35,31,43,45,39)$ $9, 9, 9, 9, 3, 3, 3$ $2880$ $9$ $( 1, 7,14,20,28,18,31,29,34)( 2,21,33,41,12,32, 8,15,45)( 3,16, 9,10,35,11,44, 19,42)( 4,37,30,23,22,26,43,38,27)( 5, 6,40)(13,24,39)(17,36,25)$ $9, 9, 9, 9, 3, 3, 3$ $2880$ $9$ $( 1,14,28,31,34, 7,20,18,29)( 2,33,12, 8,45,21,41,32,15)( 3, 9,35,44,42,16,10, 11,19)( 4,30,22,43,27,37,23,26,38)( 5,40, 6)(13,39,24)(17,25,36)$

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