Properties

Label 36T48
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times D_9$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $48$
Group :  $C_2^2\times D_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,3)(5,36)(6,35)(7,34)(8,33)(9,29)(10,30)(11,32)(12,31)(13,26)(14,25)(15,27)(16,28)(17,23)(18,24)(19,21)(20,22), (1,29,24,13,5,35,25,17,10,3,31,22,16,7,33,27,19,11)(2,30,23,14,6,36,26,18,9,4,32,21,15,8,34,28,20,12), (1,19)(2,20)(3,17)(4,18)(5,16)(6,15)(7,13)(8,14)(21,36)(22,35)(23,34)(24,33)(25,31)(26,32)(27,29)(28,30)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
12:  $D_{6}$ x 3
18:  $D_{9}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 9: $D_{9}$

Degree 12: $S_3 \times C_2^2$

Degree 18: $D_{18}$ x 3

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, 1, 1, 1, 1 $ $9$ $2$ $( 5,33)( 6,34)( 7,35)( 8,36)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,26) (16,25)(17,22)(18,21)(19,24)(20,23)$
$ 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 $ $9$ $2$ $( 1, 2)( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,31)(10,32)(11,30)(12,29)(13,28) (14,27)(15,25)(16,26)(17,21)(18,22)(19,23)(20,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5,35)( 6,36)( 7,33)( 8,34)( 9,30)(10,29)(11,31)(12,32)(13,25) (14,26)(15,28)(16,27)(17,24)(18,23)(19,22)(20,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 3)( 5,36)( 6,35)( 7,34)( 8,33)( 9,29)(10,30)(11,32)(12,31)(13,26) (14,25)(15,27)(16,28)(17,23)(18,24)(19,21)(20,22)$
$ 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, 17]
Character table: Data not available.