Properties

Label 36T30
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_2^2:C_9$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $30$
Group :  $C_2\times C_2^2:C_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,27,11,16,34,5,23,31,2,18,28,12,15,33,6,24,32)(3,20,26,10,14,36,7,22,29,4,19,25,9,13,35,8,21,30), (1,16,30,11,23,26,5,18,36,2,15,29,12,24,25,6,17,35)(3,14,32,10,22,27,7,19,34,4,13,31,9,21,28,8,20,33)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
9:  $C_9$
12:  $A_4$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: $C_9$

Degree 12: $A_4\times C_2$

Degree 18: $C_2^2 : C_9$, 18T26

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 16]
Character table: Data not available.