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Properties

Label 15T72
Order \(20160\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_8$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $72$
Group :  $A_8$
CHM label :  $L(15)=A_{8}(15)=PSL(4,2)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,5)(2,7)(3,6)(4,15)(8,9)(12,13), (1,2,15)(4,5,6)(8,9,10)(12,13,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

8T49, 15T72, 28T433, 35T36

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 7, 1 $ $2880$ $7$ $( 1, 4, 6, 9, 3,11,12)( 2,10,13,15, 5, 7, 8)$
$ 7, 7, 1 $ $2880$ $7$ $( 1,12,11, 3, 9, 6, 4)( 2, 8, 7, 5,15,13,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $210$ $2$ $( 1, 6)( 3, 9)( 4,14)( 7,10)(11,12)(13,15)$
$ 4, 4, 4, 2, 1 $ $2520$ $4$ $( 1,11, 6,12)( 2, 5)( 3,13, 9,15)( 4,10,14, 7)$
$ 3, 3, 3, 3, 1, 1, 1 $ $1120$ $3$ $( 1,12,15)( 2, 3,14)( 4, 5, 8)( 6, 7,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 7)( 6,15)(10,12)(11,13)$
$ 4, 4, 2, 2, 1, 1, 1 $ $1260$ $4$ $( 1,15,11,10)( 2, 8)( 4,14)( 6,13,12, 7)$
$ 3, 3, 3, 3, 3 $ $112$ $3$ $( 1, 4,10)( 2,14, 3)( 5, 6,12)( 7,15, 8)( 9,11,13)$
$ 6, 6, 3 $ $1680$ $6$ $( 1, 5, 4, 6,10,12)( 2,11,14,13, 3, 9)( 7, 8,15)$
$ 5, 5, 5 $ $1344$ $5$ $( 1, 8, 9,12, 3)( 2, 4, 7,11, 5)( 6,14,10,15,13)$
$ 15 $ $1344$ $15$ $( 1,11,14, 8, 5,10, 9, 2,15,12, 4,13, 3, 7, 6)$
$ 15 $ $1344$ $15$ $( 1,13, 2, 8, 6, 4, 9,14, 7,12,10,11, 3,15, 5)$
$ 6, 3, 3, 2, 1 $ $3360$ $6$ $( 1,11, 3)( 2,12, 8, 4,10,14)( 5, 7,13)( 6,15)$

Group invariants

Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  6  6  1  1  2  5  2  .  .  .   .   .  3  4
      3  2  1  2  1  2  1  1  .  .  1   1   1  .  .
      5  1  .  .  .  1  .  .  .  .  1   1   1  .  .
      7  1  .  .  .  .  .  .  1  1  .   .   .  .  .

        1a 2a 3a 6a 3b 2b 6b 7a 7b 5a 15a 15b 4a 4b
     2P 1a 1a 3a 3a 3b 1a 3b 7a 7b 5a 15a 15b 2b 2a
     3P 1a 2a 1a 2a 1a 2b 2b 7b 7a 5a  5a  5a 4a 4b
     5P 1a 2a 3a 6a 3b 2b 6b 7b 7a 1a  3b  3b 4a 4b
     7P 1a 2a 3a 6a 3b 2b 6b 1a 1a 5a 15b 15a 4a 4b
    11P 1a 2a 3a 6a 3b 2b 6b 7a 7b 5a 15b 15a 4a 4b
    13P 1a 2a 3a 6a 3b 2b 6b 7b 7a 5a 15b 15a 4a 4b

X.1      1  1  1  1  1  1  1  1  1  1   1   1  1  1
X.2      7 -1  1 -1  4  3  .  .  .  2  -1  -1  1 -1
X.3     14  6  2  . -1  2 -1  .  . -1  -1  -1  .  2
X.4     20  4 -1  1  5  4  1 -1 -1  .   .   .  .  .
X.5     21 -3  .  .  6  1 -2  .  .  1   1   1 -1  1
X.6     21 -3  .  . -3  1  1  .  .  1   B  /B -1  1
X.7     21 -3  .  . -3  1  1  .  .  1  /B   B -1  1
X.8     28 -4  1 -1  1  4  1  .  . -2   1   1  .  .
X.9     35  3  2  .  5 -5  1  .  .  .   .   . -1 -1
X.10    45 -3  .  .  . -3  .  A /A  .   .   .  1  1
X.11    45 -3  .  .  . -3  . /A  A  .   .   .  1  1
X.12    56  8 -1 -1 -4  .  .  .  .  1   1   1  .  .
X.13    64  . -2  .  4  .  .  1  1 -1  -1  -1  .  .
X.14    70 -2  1  1 -5  2 -1  .  .  .   .   .  . -2

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
B = -E(15)-E(15)^2-E(15)^4-E(15)^8
  = (-1-Sqrt(-15))/2 = -1-b15