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Properties

Label 30T46
Order \(180\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_3\times C_{15}):C_4$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $46$
Group :  $(C_3\times C_{15}):C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,5,14,8,2,10,6,15,9,3,11,4,13,7)(16,25,20,30,23,18,27,19,29,22,17,26,21,28,24), (1,20,6,28)(2,21,5,30)(3,19,4,29)(7,22,15,27)(8,23,14,26)(9,24,13,25)(10,18,12,17)(11,16)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $F_5$

Degree 6: $C_3^2:C_4$

Degree 10: $F_5$

Degree 15: None

Low degree siblings

30T46, 45T27

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 2, 3)( 4,14)( 5,13)( 6,15)( 7,12)( 8,11)( 9,10)(16,28)(17,30)(18,29)(19,27) (20,26)(21,25)(22,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,19,24,27,28,18,21,23,26, 30,17,20,22,25,29)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,20,23,27,29,17,21,24,25, 30,18,19,22,26,28)$
$ 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,21,22,27,30) (17,19,23,25,28)(18,20,24,26,29)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
$ 15, 15 $ $4$ $15$ $( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
$ 15, 15 $ $4$ $15$ $( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
$ 4, 4, 4, 4, 4, 4, 4, 2 $ $45$ $4$ $( 1,16, 7,19)( 2,17, 9,21)( 3,18, 8,20)( 4,27, 5,25)( 6,26)(10,30,15,23) (11,28,14,22)(12,29,13,24)$
$ 4, 4, 4, 4, 4, 4, 4, 2 $ $45$ $4$ $( 1,16,13,25)( 2,18,15,26)( 3,17,14,27)( 4,22,12,19)( 5,24,11,20)( 6,23,10,21) ( 7,29)( 8,28, 9,30)$

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [180, 25]
Character table:    
      2  2  .  2  .   .   .  .   .   .   .   .   .   .  2  2
      3  2  2  .  2   2   2  2   2   2   2   2   2   2  .  .
      5  1  1  .  1   1   1  1   1   1   1   1   1   1  .  .

        1a 3a 2a 3b 15a 15b 5a 15c 15d 15e 15f 15g 15h 4a 4b
     2P 1a 3a 1a 3b 15e 15h 5a 15d 15g 15b 15c 15f 15a 2a 2a
     3P 1a 1a 2a 1a  5a  5a 5a  5a  5a  5a  5a  5a  5a 4b 4a
     5P 1a 3a 2a 3b  3a  3a 1a  3b  3b  3a  3b  3b  3a 4a 4b
     7P 1a 3a 2a 3b 15h 15e 5a 15f 15c 15a 15g 15d 15b 4b 4a
    11P 1a 3a 2a 3b 15b 15a 5a 15g 15f 15h 15d 15c 15e 4b 4a
    13P 1a 3a 2a 3b 15e 15h 5a 15d 15g 15b 15c 15f 15a 4a 4b

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

A = -E(15)+E(15)^2-E(15)^4-E(15)^11+E(15)^13-E(15)^14
B = -E(15)^2+E(15)^4-E(15)^7-E(15)^8+E(15)^11-E(15)^13
C = E(15)-E(15)^2-E(15)^7-E(15)^8-E(15)^13+E(15)^14
D = -E(15)-E(15)^4+E(15)^7+E(15)^8-E(15)^11-E(15)^14
E = E(15)+E(15)^2+E(15)^13+E(15)^14
F = E(15)^2+E(15)^4+E(15)^11+E(15)^13
G = E(15)^4+E(15)^7+E(15)^8+E(15)^11
H = E(15)+E(15)^7+E(15)^8+E(15)^14
I = -E(4)
  = -Sqrt(-1) = -i