Properties

Label 36T46
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_9:D_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $46$
Group :  $C_9:D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,2,17)(3,19,4,20)(5,14,6,13)(7,16,8,15)(9,11,10,12)(21,34,22,33)(23,35,24,36)(25,30,26,29)(27,31,28,32), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11)(17,36)(18,35)(19,33)(20,34)(21,29)(22,30)(23,32)(24,31)(27,28)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Degree 12: $(C_6\times C_2):C_2$

Degree 18: $D_{18}$

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 8]
Character table: Data not available.