# Properties

 Label 27T993 Order $$25920$$ n $$27$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSp(4,3)$

## Group action invariants

 Degree $n$ : $27$ Transitive number $t$ : $993$ Group : $\PSp(4,3)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,20,25)(3,17,15,22)(4,8)(5,18,24,7)(6,19,26,9)(10,11,27,21)(13,14)(16,23), (1,16,24,4,5,8,7,21,25)(2,9,17,14,22,19,11,26,6)(3,20,10,12,27,18,15,13,23) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 3: None

Degree 9: None

## Low degree siblings

36T12781, 40T14344, 40T14345, 45T666

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

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