Properties

Label 36T25
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2:D_9$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $25$
Group :  $C_2^2:D_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,2,21)(3,23)(4,24)(5,20,6,19)(7,18,8,17)(9,16,10,15)(11,14)(12,13)(25,34)(26,33)(27,35,28,36)(29,31,30,32), (1,33,32,27,24,20,16,11,7)(2,34,31,28,23,19,15,12,8)(3,35,30,25,22,17,13,10,5)(4,36,29,26,21,18,14,9,6)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
18:  $D_{9}$
24:  $S_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$, $S_4$, $S_4$

Degree 9: $D_{9}$

Degree 12: $S_4$

Degree 18: $D_9$, 18T38, 18T39

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(19,20)(21,22)(23,24)(25,26)(31,32)(33,34) (35,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,31)(10,32)(11,30)(12,29)(13,27) (14,28)(15,26)(16,25)(17,24)(18,23)(19,21)(20,22)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2 $ $18$ $4$ $( 1, 3, 2, 4)( 5,33, 6,34)( 7,36)( 8,35)( 9,32)(10,31)(11,29,12,30) (13,28,14,27)(15,26,16,25)(17,24,18,23)(19,22)(20,21)$
$ 9, 9, 9, 9 $ $8$ $9$ $( 1, 7,11,16,20,24,27,32,33)( 2, 8,12,15,19,23,28,31,34)( 3, 5,10,13,17,22,25, 30,35)( 4, 6, 9,14,18,21,26,29,36)$
$ 9, 9, 9, 9 $ $8$ $9$ $( 1,11,19,27,33, 8,16,24,31)( 2,12,20,28,34, 7,15,23,32)( 3, 9,17,25,36, 5,13, 21,30)( 4,10,18,26,35, 6,14,22,29)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $6$ $6$ $( 1,15,27, 2,16,28)( 3,13,25)( 4,14,26)( 5,18,30, 6,17,29)( 7,19,32, 8,20,31) ( 9,22,36,10,21,35)(11,24,33)(12,23,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,16,27)( 2,15,28)( 3,13,25)( 4,14,26)( 5,17,30)( 6,18,29)( 7,20,32) ( 8,19,31)( 9,21,36)(10,22,35)(11,24,33)(12,23,34)$
$ 9, 9, 9, 9 $ $8$ $9$ $( 1,19,33,16,31,11,27, 8,24)( 2,20,34,15,32,12,28, 7,23)( 3,17,36,13,30, 9,25, 5,21)( 4,18,35,14,29,10,26, 6,22)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 15]
Character table:   
     2  3  3  2  2  .  .  2  2  .
     3  2  1  .  .  2  2  1  2  2

       1a 2a 2b 4a 9a 9b 6a 3a 9c
    2P 1a 1a 1a 2a 9b 9c 3a 3a 9a
    3P 1a 2a 2b 4a 3a 3a 2a 1a 3a
    5P 1a 2a 2b 4a 9c 9a 6a 3a 9b
    7P 1a 2a 2b 4a 9b 9c 6a 3a 9a

X.1     1  1  1  1  1  1  1  1  1
X.2     1  1 -1 -1  1  1  1  1  1
X.3     2  2  .  . -1 -1  2  2 -1
X.4     2  2  .  .  A  C -1 -1  B
X.5     2  2  .  .  B  A -1 -1  C
X.6     2  2  .  .  C  B -1 -1  A
X.7     3 -1 -1  1  .  . -1  3  .
X.8     3 -1  1 -1  .  . -1  3  .
X.9     6 -2  .  .  .  .  1 -3  .

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