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Properties

Label 28T70
Order \(504\)
n \(28\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,8)$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $70$
Group :  $\PSL(2,8)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,27,15,25,10,24,7,20)(2,19,8,26,17,16,6,5,3)(9,28,18,13,11,22,14,23,12), (1,28,21,4,25,18,10)(2,26,19,17,12,7,11)(3,24,20,16,13,14,9)(5,6,23,15,8,27,22)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

9T27, 36T712

Siblings are shown with degree $\leq 47$

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

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $63$ $2$ $( 3, 7)( 4,28)( 5,14)( 6,27)( 8,11)( 9,10)(12,13)(15,21)(16,22)(17,24)(18,20) (19,23)$
$ 9, 9, 9, 1 $ $56$ $9$ $( 2, 3,27, 8,13,12,11, 6, 7)( 4,14,19, 9,25,10,23, 5,28)(15,20,22,26,16,18,21, 24,17)$
$ 9, 9, 9, 1 $ $56$ $9$ $( 2, 6,12, 8, 3, 7,11,13,27)( 4, 5,10, 9,14,28,23,25,19)(15,24,18,26,20,17,21, 16,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $56$ $3$ $( 2, 8,11)( 3,13, 6)( 4, 9,23)( 5,14,25)( 7,27,12)(10,28,19)(15,26,21) (16,24,20)(17,22,18)$
$ 9, 9, 9, 1 $ $56$ $9$ $( 2,12, 3,11,27, 6, 8, 7,13)( 4,10,14,23,19, 5, 9,28,25)(15,18,20,21,22,24,26, 17,16)$
$ 7, 7, 7, 7 $ $72$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,21,24,17,12,23,14)( 9,26,22,25,18,13,15) (10,11,28,27,19,20,16)$
$ 7, 7, 7, 7 $ $72$ $7$ $( 1, 2,15,10, 5,20,13)( 3,23, 8, 7,21,16,12)( 4,18, 9, 6,19,14,11) (17,22,24,25,27,28,26)$
$ 7, 7, 7, 7 $ $72$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,17,14,24,23,21,12)( 9,25,15,22,13,26,18) (10,27,16,28,20,11,19)$

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 9a 9b 3a 9c 7a 7b 7c
    2P 1a 1a 9b 9c 3a 9a 7b 7c 7a
    3P 1a 2a 3a 3a 1a 3a 7c 7a 7b
    5P 1a 2a 9c 9a 3a 9b 7b 7c 7a
    7P 1a 2a 9b 9c 3a 9a 1a 1a 1a

X.1     1  1  1  1  1  1  1  1  1
X.2     7 -1  1  1 -2  1  .  .  .
X.3     7 -1  A  C  1  B  .  .  .
X.4     7 -1  B  A  1  C  .  .  .
X.5     7 -1  C  B  1  A  .  .  .
X.6     8  . -1 -1 -1 -1  1  1  1
X.7     9  1  .  .  .  .  D  F  E
X.8     9  1  .  .  .  .  E  D  F
X.9     9  1  .  .  .  .  F  E  D

A = -E(9)^4-E(9)^5
B = -E(9)^2-E(9)^7
C = E(9)^2+E(9)^4+E(9)^5+E(9)^7
D = E(7)^3+E(7)^4
E = E(7)^2+E(7)^5
F = E(7)+E(7)^6