Properties

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

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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 $\leq 10$

Subfields

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 TypeSizeOrderRepresentative
$ 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