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Properties

Label 27T30
Order \(108\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_9$

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Group action invariants

Degree $n$ :  $27$
Transitive number $t$ :  $30$
Group :  $S_3\times D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,10,3,21,12,2,20,11)(4,17,14,25,22,9,5,18,15,26,23,7,6,16,13,27,24,8), (1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2
18:  $D_{9}$
36:  $S_3^2$, $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 9: $D_{9}$, $S_3^2$

Low degree siblings

18T50, 36T86

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 16]
Character table:    
      2  2  2  2  2  1  1  1  1  .  .   1  .   1  .   1  1  1  1
      3  3  2  1  .  3  2  3  1  3  3   2  3   2  3   2  3  3  3

        1a 2a 2b 2c 3a 6a 3b 6b 3c 9a 18a 9b 18b 9c 18c 9d 9e 9f
     2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 9c  9f 9a  9d 9b  9e 9f 9d 9e
     3P 1a 2a 2b 2c 1a 2a 1a 2b 1a 3a  6a 3a  6a 3a  6a 3a 3a 3a
     5P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9b 18b 9c 18c 9a 18a 9e 9f 9d
     7P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9c 18c 9a 18a 9b 18b 9f 9d 9e
    11P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9c 18c 9a 18a 9b 18b 9f 9d 9e
    13P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9b 18b 9c 18c 9a 18a 9e 9f 9d
    17P 1a 2a 2b 2c 3a 6a 3b 6b 3c 9a 18a 9b 18b 9c 18c 9d 9e 9f

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

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