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Properties

Label 15T9
Order \(75\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2 : C_3$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $9$
Group :  $C_5^2 : C_3$
CHM label :  $[5^{2}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T9, 25T6

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 $ $1$ $1$ $()$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $3$ $5$ $( 2,14,11, 8, 5)( 3, 6, 9,12,15)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $3$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,15,12, 9, 6)$

Group invariants

Order:  $75=3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [75, 2]
Character table:    
      3  1  .  .  .  .  1  1  .  .  .  .
      5  2  2  2  2  2  .  .  2  2  2  2

        1a 5a 5b 5c 5d 3a 3b 5e 5f 5g 5h
     2P 1a 5b 5d 5a 5c 3b 3a 5f 5g 5h 5e
     3P 1a 5c 5a 5d 5b 1a 1a 5h 5e 5f 5g
     5P 1a 1a 1a 1a 1a 3b 3a 1a 1a 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  1  D /D  1  1  1  1
X.3      1  1  1  1  1 /D  D  1  1  1  1
X.4      3  A /B  B /A  .  .  C *C  C *C
X.5      3  B  A /A /B  .  . *C  C *C  C
X.6      3 /B /A  A  B  .  . *C  C *C  C
X.7      3 /A  B /B  A  .  .  C *C  C *C
X.8      3  C *C *C  C  .  .  B  A /B /A
X.9      3  C *C *C  C  .  . /B /A  B  A
X.10     3 *C  C  C *C  .  . /A  B  A /B
X.11     3 *C  C  C *C  .  .  A /B /A  B

A = 2*E(5)+E(5)^3
B = 2*E(5)^3+E(5)^4
C = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
D = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3