Properties

Label 24T576
Degree $24$
Order $240$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_4.A_5$

Related objects

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Group action invariants

Degree $n$:  $24$
Transitive number $t$:  $576$
Group:  $C_4.A_5$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $4$
Generators:  (1,13,20,16,5)(2,14,19,15,6)(3,24,12,21,17)(4,23,11,22,18), (1,8,5,23,10,4,2,7,6,24,9,3)(11,16,20,14,18,22,12,15,19,13,17,21)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$
$120$:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $\PSL(2,5)$

Degree 8: None

Degree 12: 12T76

Low degree siblings

24T576, 40T188 x 2

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, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $30$ $2$ $( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(11,15)(12,16)(13,18)(14,17)$
$ 5, 5, 5, 5, 1, 1, 1, 1 $ $12$ $5$ $( 3, 8,11,16,19)( 4, 7,12,15,20)( 5,10,14,18,21)( 6, 9,13,17,22)$
$ 5, 5, 5, 5, 1, 1, 1, 1 $ $12$ $5$ $( 3,11,19, 8,16)( 4,12,20, 7,15)( 5,14,21,10,18)( 6,13,22, 9,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 10, 10, 2, 2 $ $12$ $10$ $( 1, 2)( 3, 7,11,15,19, 4, 8,12,16,20)( 5, 9,14,17,21, 6,10,13,18,22)(23,24)$
$ 10, 10, 2, 2 $ $12$ $10$ $( 1, 2)( 3,12,19, 7,16, 4,11,20, 8,15)( 5,13,21, 9,18, 6,14,22,10,17)(23,24)$
$ 4, 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 3, 2, 4)( 5,24, 6,23)( 7, 9, 8,10)(11,16,12,15)(13,17,14,18)(19,22,20,21)$
$ 12, 12 $ $20$ $12$ $( 1, 3, 9,24, 6, 7, 2, 4,10,23, 5, 8)(11,21,17,13,19,15,12,22,18,14,20,16)$
$ 20, 4 $ $12$ $20$ $( 1, 3,11, 9,18,23, 5,14, 8,15, 2, 4,12,10,17,24, 6,13, 7,16)(19,21,20,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 3,12)( 2, 4,11)( 5,13,23)( 6,14,24)( 7,21,18)( 8,22,17)( 9,20,15) (10,19,16)$
$ 20, 4 $ $12$ $20$ $( 1, 3,15,22,13,24, 6,18,20,12, 2, 4,16,21,14,23, 5,17,19,11)( 7,10, 8, 9)$
$ 6, 6, 6, 6 $ $20$ $6$ $( 1, 3,16, 2, 4,15)( 5,18,24, 6,17,23)( 7,11,22, 8,12,21)( 9,14,19,10,13,20)$
$ 12, 12 $ $20$ $12$ $( 1, 3,21,23, 5,20, 2, 4,22,24, 6,19)( 7,12,16, 9,13,18, 8,11,15,10,14,17)$
$ 20, 4 $ $12$ $20$ $( 1, 5, 9,16,19,24, 3, 7,17,22, 2, 6,10,15,20,23, 4, 8,18,21)(11,13,12,14)$
$ 20, 4 $ $12$ $20$ $( 1, 5,21,11, 7,23, 4,20,14, 9, 2, 6,22,12, 8,24, 3,19,13,10)(15,17,16,18)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,23, 2,24)( 3, 5, 4, 6)( 7, 9, 8,10)(11,14,12,13)(15,17,16,18)(19,21,20,22)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,24, 2,23)( 3, 6, 4, 5)( 7,10, 8, 9)(11,13,12,14)(15,18,16,17)(19,22,20,21)$

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  [240, 93]
Character table:   
      2  4  3   2   2  4   2   2  3   2   2  2   2  2   2   2   2  4  4
      3  1  .   .   .  1   .   .  .   1   .  1   .  1   1   .   .  1  1
      5  1  .   1   1  1   1   1  .   .   1  .   1  .   .   1   1  1  1

        1a 2a  5a  5b 2b 10a 10b 4a 12a 20a 3a 20b 6a 12b 20c 20d 4b 4c
     2P 1a 1a  5b  5a 1a  5b  5a 2b  6a 10b 3a 10b 3a  6a 10a 10a 2b 2b
     3P 1a 2a  5b  5a 2b 10b 10a 4a  4c 20c 1a 20d 2b  4b 20a 20b 4c 4b
     5P 1a 2a  1a  1a 2b  2b  2b 4a 12a  4b 3a  4c 6a 12b  4c  4b 4b 4c
     7P 1a 2a  5b  5a 2b 10b 10a 4a 12b 20c 3a 20d 6a 12a 20a 20b 4c 4b
    11P 1a 2a  5a  5b 2b 10a 10b 4a 12b 20b 3a 20a 6a 12a 20d 20c 4c 4b
    13P 1a 2a  5b  5a 2b 10b 10a 4a 12a 20d 3a 20c 6a 12b 20b 20a 4b 4c
    17P 1a 2a  5b  5a 2b 10b 10a 4a 12a 20d 3a 20c 6a 12b 20b 20a 4b 4c
    19P 1a 2a  5a  5b 2b 10a 10b 4a 12b 20b 3a 20a 6a 12a 20d 20c 4c 4b

X.1      1  1   1   1  1   1   1  1   1   1  1   1  1   1   1   1  1  1
X.2      1 -1   1   1  1   1   1  1  -1  -1  1  -1  1  -1  -1  -1 -1 -1
X.3      2  .   A  *A -2  -A -*A  .   B   C -1  -C  1  -B  -D   D  E -E
X.4      2  .  *A   A -2 -*A  -A  .   B   D -1  -D  1  -B  -C   C  E -E
X.5      2  .   A  *A -2  -A -*A  .  -B  -C -1   C  1   B   D  -D -E  E
X.6      2  .  *A   A -2 -*A  -A  .  -B  -D -1   D  1   B   C  -C -E  E
X.7      3 -1 -*A  -A  3 -*A  -A -1   . -*A  . -*A  .   .  -A  -A  3  3
X.8      3 -1  -A -*A  3  -A -*A -1   .  -A  .  -A  .   . -*A -*A  3  3
X.9      3  1 -*A  -A  3 -*A  -A -1   .  *A  .  *A  .   .   A   A -3 -3
X.10     3  1  -A -*A  3  -A -*A -1   .   A  .   A  .   .  *A  *A -3 -3
X.11     4  .  -1  -1  4  -1  -1  .   1  -1  1  -1  1   1  -1  -1  4  4
X.12     4  .  -1  -1  4  -1  -1  .  -1   1  1   1  1  -1   1   1 -4 -4
X.13     4  .  -1  -1 -4   1   1  .  -B   B  1  -B -1   B  -B   B  F -F
X.14     4  .  -1  -1 -4   1   1  .   B  -B  1   B -1  -B   B  -B -F  F
X.15     5  1   .   .  5   .   .  1  -1   . -1   . -1  -1   .   .  5  5
X.16     5 -1   .   .  5   .   .  1   1   . -1   . -1   1   .   . -5 -5
X.17     6  .   1   1 -6  -1  -1  .   .  -B  .   B  .   .   B  -B  G -G
X.18     6  .   1   1 -6  -1  -1  .   .   B  .  -B  .   .  -B   B -G  G

A = E(5)+E(5)^4
  = (-1+Sqrt(5))/2 = b5
B = E(4)
  = Sqrt(-1) = i
C = -E(20)-E(20)^9
D = -E(20)^13-E(20)^17
E = -2*E(4)
  = -2*Sqrt(-1) = -2i
F = -4*E(4)
  = -4*Sqrt(-1) = -4i
G = -6*E(4)
  = -6*Sqrt(-1) = -6i