# Properties

 Label 36T6815 Order $$6048$$ n $$36$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSU(3,3)$

## Group action invariants

 Degree $n$ : $36$ Transitive number $t$ : $6815$ Group : $\PSU(3,3)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,29)(26,28,30)(31,35,34)(32,36,33), (5,6)(8,9)(10,11)(13,14)(17,18)(19,20)(23,24)(27,29)(28,30)(31,33)(32,34)(35,36), (3,5,4,6)(7,12)(8,10,11,9)(13,14)(16,21)(17,19,20,18)(23,24)(25,35,26,36)(27,32,33,30)(28,31,34,29), (1,2,22,15)(3,17,26,11,4,20,25,8)(5,18,36,10,6,19,35,9)(7,24,12,23)(13,16,14,21)(27,31,32,28,33,29,30,34) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $63$ $2$ $( 1,33)( 2,32)( 3,36)( 5,18)( 6, 8)( 9,17)(11,25)(12,23)(13,21)(15,30)(19,26) (22,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $56$ $3$ $( 1, 9,15)( 2,19,23)( 3,25, 8)( 4,34,16)( 5,27,21)( 6,36,11)( 7,31,35) (10,14,24)(12,32,26)(13,18,22)(17,30,33)(20,29,28)$ $4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2$ $63$ $4$ $( 1, 8,33, 6)( 2, 5,32,18)( 3,17,36, 9)( 4,35)( 7,34)(10,20)(11,15,25,30) (12,13,23,21)(14,29)(16,31)(19,27,26,22)(24,28)$ $4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2$ $63$ $4$ $( 1, 6,33, 8)( 2,18,32, 5)( 3, 9,36,17)( 4,35)( 7,34)(10,20)(11,30,25,15) (12,21,23,13)(14,29)(16,31)(19,22,26,27)(24,28)$ $6, 6, 6, 6, 3, 3, 3, 3$ $504$ $6$ $( 1,30, 9,33,15,17)( 2,12,19,32,23,26)( 3, 6,25,36, 8,11)( 4,16,34) ( 5,13,27,18,21,22)( 7,35,31)(10,24,14)(20,28,29)$ $12, 12, 6, 6$ $504$ $12$ $( 1,36,30, 8, 9,11,33, 3,15, 6,17,25)( 2,22,12, 5,19,13,32,27,23,18,26,21) ( 4, 7,16,35,34,31)(10,29,24,20,14,28)$ $12, 12, 6, 6$ $504$ $12$ $( 1, 3,30, 6, 9,25,33,36,15, 8,17,11)( 2,27,12,18,19,21,32,22,23, 5,26,13) ( 4, 7,16,35,34,31)(10,29,24,20,14,28)$ $8, 8, 8, 4, 4, 4$ $756$ $8$ $( 1, 7,36, 5)( 2,14,16,17,12, 3,22,28)( 4,33,32, 8,13,29,19,10)( 6,27,15,31) ( 9,35,11,21)(18,20,23,24,34,30,26,25)$ $8, 8, 8, 4, 4, 4$ $756$ $8$ $( 1, 5,36, 7)( 2,28,22, 3,12,17,16,14)( 4,10,19,29,13, 8,32,33)( 6,31,15,27) ( 9,21,11,35)(18,25,26,30,34,24,23,20)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $672$ $3$ $( 1,29,18)( 2, 7,26)( 4,21,22)( 5,36,32)( 6,13,10)( 8,28, 9)(11,24,35) (12,19,33)(14,34,23)(15,20,31)(16,27,17)$ $7, 7, 7, 7, 7, 1$ $864$ $7$ $( 1,33, 3,14, 8, 7,17)( 2,10,25,27,34,13,36)( 4,30, 9,23,29,22, 5) ( 6,18,28,20,35,31,11)(12,32,16,24,15,21,26)$ $7, 7, 7, 7, 7, 1$ $864$ $7$ $( 1,17, 7, 8,14, 3,33)( 2,36,13,34,27,25,10)( 4, 5,22,29,23, 9,30) ( 6,11,31,35,20,28,18)(12,26,21,15,24,16,32)$ $4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1$ $378$ $4$ $( 1,12,25, 4)( 2,13,28,33)( 3,32)( 5,30,20, 7)( 6,22,31,10)( 8,16)( 9,26) (11,14,21,19)(15,34)(17,24,29,36)$

## Group invariants

 Order: $6048=2^{5} \cdot 3^{3} \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 5 5 2 5 5 2 2 2 3 3 . . . 4 3 3 1 3 1 1 1 1 1 . . . . 2 . 7 1 . . . . . . . . . 1 1 . . 1a 2a 3a 4a 4b 6a 12a 12b 8a 8b 7a 7b 3b 4c 2P 1a 1a 3a 2a 2a 3a 6a 6a 4b 4a 7a 7b 3b 2a 3P 1a 2a 1a 4b 4a 2a 4a 4b 8b 8a 7b 7a 1a 4c 5P 1a 2a 3a 4a 4b 6a 12a 12b 8a 8b 7b 7a 3b 4c 7P 1a 2a 3a 4b 4a 6a 12b 12a 8b 8a 1a 1a 3b 4c 11P 1a 2a 3a 4b 4a 6a 12b 12a 8b 8a 7a 7b 3b 4c X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 6 -2 -3 -2 -2 1 1 1 . . -1 -1 . 2 X.3 7 -1 -2 3 3 2 . . -1 -1 . . 1 -1 X.4 7 3 -2 A /A . D /D E -E . . 1 1 X.5 7 3 -2 /A A . /D D -E E . . 1 1 X.6 14 -2 5 2 2 1 -1 -1 . . . . -1 2 X.7 21 5 3 1 1 -1 1 1 -1 -1 . . . 1 X.8 21 1 3 B /B 1 E -E -E E . . . -1 X.9 21 1 3 /B B 1 -E E E -E . . . -1 X.10 27 3 . 3 3 . . . 1 1 -1 -1 . -1 X.11 28 -4 1 C -C -1 -E E . . . . 1 . X.12 28 -4 1 -C C -1 E -E . . . . 1 . X.13 32 . -4 . . . . . . . F /F -1 . X.14 32 . -4 . . . . . . . /F F -1 . A = -1-2*E(4) = -1-2*Sqrt(-1) = -1-2i B = -3-2*E(4) = -3-2*Sqrt(-1) = -3-2i C = -4*E(4) = -4*Sqrt(-1) = -4i D = -1-E(4) = -1-Sqrt(-1) = -1-i E = -E(4) = -Sqrt(-1) = -i F = -E(7)-E(7)^2-E(7)^4 = (1-Sqrt(-7))/2 = -b7