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

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