# Properties

 Label 36T10 Order $$36$$ n $$36$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{18}$

## Group action invariants

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

## Low degree resolvents

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

Degree 9: $D_{9}$

Degree 12: $D_6$

Degree 18: $D_9$, $D_{18}$ x 2

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

## Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [36, 4]
 Character table:  2 2 2 2 2 1 1 1 1 1 1 1 1 3 2 2 . . 2 2 2 2 2 2 2 2 1a 2a 2b 2c 9a 18a 18b 9b 6a 3a 9c 18c 2P 1a 1a 1a 1a 9b 9b 9c 9c 3a 3a 9a 9a 3P 1a 2a 2b 2c 3a 6a 6a 3a 2a 1a 3a 6a 5P 1a 2a 2b 2c 9c 18c 18a 9a 6a 3a 9b 18b 7P 1a 2a 2b 2c 9b 18b 18c 9c 6a 3a 9a 18a 11P 1a 2a 2b 2c 9b 18b 18c 9c 6a 3a 9a 18a 13P 1a 2a 2b 2c 9c 18c 18a 9a 6a 3a 9b 18b 17P 1a 2a 2b 2c 9a 18a 18b 9b 6a 3a 9c 18c X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 X.3 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 X.4 1 1 -1 -1 1 1 1 1 1 1 1 1 X.5 2 2 . . -1 -1 -1 -1 2 2 -1 -1 X.6 2 -2 . . -1 1 1 -1 -2 2 -1 1 X.7 2 2 . . A A B B -1 -1 C C X.8 2 2 . . B B C C -1 -1 A A X.9 2 2 . . C C A A -1 -1 B B X.10 2 -2 . . A -A -B B 1 -1 C -C X.11 2 -2 . . B -B -C C 1 -1 A -A X.12 2 -2 . . C -C -A A 1 -1 B -B A = E(9)^2+E(9)^7 B = E(9)^4+E(9)^5 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7