Properties

Label 36T24
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 $t$ :  $24$
Group :  $C_9:D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,7,27,12,31,16,33,19,2,24,8,28,11,32,15,34,20)(3,22,5,25,10,30,13,35,17)(4,21,6,26,9,29,14,36,18), (1,5,2,6)(3,8,4,7)(9,34,10,33)(11,36,12,35)(13,31,14,32)(15,29,16,30)(17,27,18,28)(19,25,20,26)(21,24,22,23)
$|\Aut(F/K)|$:  $18$

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: $S_3$

Degree 9: $D_{9}$

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

Degree 18: $D_9$

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

Group invariants

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