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Properties

Label 36T185
Order \(216\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_2\times C_{18}):C_6$

Related objects

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $185$
Group :  $(C_2\times C_{18}):C_6$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,2,23)(3,21,4,22)(5,19,6,20)(7,18,8,17)(9,15,10,16)(11,13,12,14)(25,36,26,35)(27,33,28,34)(29,32,30,31), (1,31,33,14,8,9,25,19,21)(2,32,34,13,7,10,26,20,22)(3,29,35,15,5,11,27,17,23,4,30,36,16,6,12,28,18,24)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $D_{4}$
12:  $D_{6}$
18:  $S_3\times C_3$
54:  $(C_9:C_3):C_2$

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: $(C_9:C_3):C_2$

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

Degree 18: 18T18

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

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [216, 62]
Character table: Data not available.