Properties

Label 27T8
Order \(54\)
n \(27\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{27}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
18:  $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $D_{9}$

Low degree siblings

There are no siblings 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, 2, 2, 2, 2, 1 $ $27$ $2$ $( 2, 3)( 4,27)( 5,26)( 6,25)( 7,23)( 8,22)( 9,24)(10,20)(11,19)(12,21)(13,18) (14,17)(15,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 27 $ $2$ $27$ $( 1, 4, 8,12,15,18,20,24,26, 3, 6, 7,11,14,17,19,23,25, 2, 5, 9,10,13,16,21, 22,27)$
$ 27 $ $2$ $27$ $( 1, 5, 7,12,13,17,20,22,25, 3, 4, 9,11,15,16,19,24,27, 2, 6, 8,10,14,18,21, 23,26)$
$ 27 $ $2$ $27$ $( 1, 6, 9,12,14,16,20,23,27, 3, 5, 8,11,13,18,19,22,26, 2, 4, 7,10,15,17,21, 24,25)$
$ 27 $ $2$ $27$ $( 1, 7,13,20,25, 4,11,16,24, 2, 8,14,21,26, 5,12,17,22, 3, 9,15,19,27, 6,10, 18,23)$
$ 27 $ $2$ $27$ $( 1, 8,15,20,26, 6,11,17,23, 2, 9,13,21,27, 4,12,18,24, 3, 7,14,19,25, 5,10, 16,22)$
$ 27 $ $2$ $27$ $( 1, 9,14,20,27, 5,11,18,22, 2, 7,15,21,25, 6,12,16,23, 3, 8,13,19,26, 4,10, 17,24)$
$ 9, 9, 9 $ $2$ $9$ $( 1,10,19, 3,12,21, 2,11,20)( 4,13,23, 6,15,22, 5,14,24)( 7,18,27, 9,17,26, 8, 16,25)$
$ 9, 9, 9 $ $2$ $9$ $( 1,11,21, 3,10,20, 2,12,19)( 4,14,22, 6,13,24, 5,15,23)( 7,16,26, 9,18,25, 8, 17,27)$
$ 9, 9, 9 $ $2$ $9$ $( 1,12,20, 3,11,19, 2,10,21)( 4,15,24, 6,14,23, 5,13,22)( 7,17,25, 9,16,27, 8, 18,26)$
$ 27 $ $2$ $27$ $( 1,13,25,11,24, 8,21, 5,17, 3,15,27,10,23, 7,20, 4,16, 2,14,26,12,22, 9,19, 6,18)$
$ 27 $ $2$ $27$ $( 1,14,27,11,22, 7,21, 6,16, 3,13,26,10,24, 9,20, 5,18, 2,15,25,12,23, 8,19, 4,17)$
$ 27 $ $2$ $27$ $( 1,15,26,11,23, 9,21, 4,18, 3,14,25,10,22, 8,20, 6,17, 2,13,27,12,24, 7,19, 5,16)$

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 1]
Character table:   
      2  1  1  .   .   .   .   .   .   .  .  .  .   .   .   .
      3  3  .  3   3   3   3   3   3   3  3  3  3   3   3   3

        1a 2a 3a 27a 27b 27c 27d 27e 27f 9a 9b 9c 27g 27h 27i
     2P 1a 1a 3a 27e 27d 27f 27g 27i 27h 9b 9c 9a 27c 27a 27b
     3P 1a 2a 1a  9c  9c  9c  9a  9a  9a 3a 3a 3a  9b  9b  9b
     5P 1a 2a 3a 27g 27h 27i 27a 27c 27b 9c 9a 9b 27e 27d 27f
     7P 1a 2a 3a 27f 27e 27d 27i 27h 27g 9b 9c 9a 27b 27c 27a
    11P 1a 2a 3a 27d 27f 27e 27h 27g 27i 9b 9c 9a 27a 27b 27c
    13P 1a 2a 3a 27h 27i 27g 27b 27a 27c 9c 9a 9b 27d 27f 27e
    17P 1a 2a 3a 27c 27a 27b 27e 27f 27d 9a 9b 9c 27i 27g 27h
    19P 1a 2a 3a 27b 27c 27a 27f 27d 27e 9a 9b 9c 27h 27i 27g
    23P 1a 2a 3a 27i 27g 27h 27c 27b 27a 9c 9a 9b 27f 27e 27d

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

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 = E(27)^13+E(27)^14
E = E(27)^5+E(27)^22
F = -E(27)^5-E(27)^13-E(27)^14-E(27)^22
G = E(27)^11+E(27)^16
H = -E(27)^7-E(27)^11-E(27)^16-E(27)^20
I = E(27)^7+E(27)^20
J = -E(27)^8-E(27)^10-E(27)^17-E(27)^19
K = E(27)^10+E(27)^17
L = E(27)^8+E(27)^19