# Properties

 Label 36T712 Order $$504$$ n $$36$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(2,8)$

## Group action invariants

 Degree $n$ : $36$ Transitive number $t$ : $712$ Group : $\PSL(2,8)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,6,17,19,8,4)(3,10,23,12,28,25,13)(5,15,9,22,20,14,18)(7,16,31,30,29,21,32)(11,26,34,35,27,33,24), (1,4)(2,8)(3,11)(5,14)(6,19)(7,21)(9,22)(10,24)(12,27)(13,26)(15,20)(16,29)(23,33)(25,34)(28,35)(30,31), (1,3)(2,7)(4,13)(5,12)(6,18)(8,9)(10,14)(11,26)(16,23)(17,31)(20,32)(21,22)(24,27)(25,29)(33,34)(35,36) $|\Aut(F/K)|$: $1$

## 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

9T27

Siblings are shown 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $63$ $2$ $( 2, 4)( 3,26)( 5,18)( 6, 8)( 7,32)( 9,20)(10,11)(12,33)(13,34)(14,15)(16,21) (17,19)(23,24)(25,35)(27,28)(29,31)$ $7, 7, 7, 7, 7, 1$ $72$ $7$ $( 2,15,27,25,31,18,21)( 3,26,12,19,30,17,33)( 4,16, 5,29,35,28,14) ( 6,24,36,23, 8,20, 9)( 7,11,13,22,34,10,32)$ $7, 7, 7, 7, 7, 1$ $72$ $7$ $( 2,18,25,15,21,31,27)( 3,17,19,26,33,30,12)( 4,28,29,16,14,35, 5) ( 6,20,23,24, 9, 8,36)( 7,10,22,11,32,34,13)$ $7, 7, 7, 7, 7, 1$ $72$ $7$ $( 2,25,21,27,18,15,31)( 3,19,33,12,17,26,30)( 4,29,14, 5,28,16,35) ( 6,23, 9,36,20,24, 8)( 7,22,32,13,10,11,34)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $56$ $3$ $( 1, 2, 5)( 3,19,34)( 4,25, 9)( 6,14,27)( 7,23,11)( 8,36,17)(10,20,12) (13,16,18)(15,32,35)(21,30,28)(22,26,24)(29,31,33)$ $9, 9, 9, 9$ $56$ $9$ $( 1, 2, 9,17,24,36,12, 8,14)( 3,34,23, 4,31, 5,21, 6,11)( 7,26,28,18,32,16,15, 33,10)(13,20,22,35,25,30,19,29,27)$ $9, 9, 9, 9$ $56$ $9$ $( 1, 2,13,23,22, 7,20,11,35)( 3, 8,17,34,16,27, 5,25,32)( 4,21,33,28,31, 6,12, 19,24)( 9,36,14,15,30,10,26,29,18)$ $9, 9, 9, 9$ $56$ $9$ $( 1, 2,30,34,26,19,10, 3,29)( 4,27,32,20,36,22,16,25, 6)( 5,18,33, 8,11,23,17, 28,15)( 7,35,31, 9,12,24,14,21,13)$

## Group invariants

 Order: $504=2^{3} \cdot 3^{2} \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: [504, 156]
 Character table:  2 3 3 . . . . . . . 3 2 . . . . 2 2 2 2 7 1 . 1 1 1 . . . . 1a 2a 7a 7b 7c 3a 9a 9b 9c 2P 1a 1a 7b 7c 7a 3a 9c 9a 9b 3P 1a 2a 7c 7a 7b 1a 3a 3a 3a 5P 1a 2a 7b 7c 7a 3a 9b 9c 9a 7P 1a 2a 1a 1a 1a 3a 9c 9a 9b X.1 1 1 1 1 1 1 1 1 1 X.2 7 -1 . . . -2 1 1 1 X.3 7 -1 . . . 1 D E F X.4 7 -1 . . . 1 E F D X.5 7 -1 . . . 1 F D E X.6 8 . 1 1 1 -1 -1 -1 -1 X.7 9 1 A C B . . . . X.8 9 1 B A C . . . . X.9 9 1 C B A . . . . A = E(7)^3+E(7)^4 B = E(7)^2+E(7)^5 C = E(7)+E(7)^6 D = -E(9)^4-E(9)^5 E = -E(9)^2-E(9)^7 F = E(9)^2+E(9)^4+E(9)^5+E(9)^7