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Properties

Label 18T273
Order \(1152\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $273$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,7)(2,18,8)(3,13,10,5,16,12,4,14,9,6,15,11), (1,18,13,6)(2,17,14,5)(3,8,16,11,4,7,15,12)(9,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
72:  $C_3^2:D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 9: $S_3^2:C_2$

Low degree siblings

8T47, 12T200, 12T201, 12T202, 12T203, 16T1292, 16T1294, 16T1295, 16T1296, 18T272, 18T274, 18T275

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)( 9,10)(15,16)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,17, 7)( 2,18, 8)( 3,16, 9)( 4,15,10)( 5,14,11)( 6,13,12)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,18, 7, 2,17, 8)( 3,16, 9)( 4,15,10)( 5,13,11, 6,14,12)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1,13, 9)( 2,14,10)( 3,17,12)( 4,18,11)( 5,15, 8)( 6,16, 7)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $12$ $4$ $( 3, 6, 4, 5)( 9,12,10,11)(13,15,14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)(17,18)$
$ 4, 4, 4, 2, 2, 1, 1 $ $36$ $4$ $( 3, 6, 4, 5)( 7, 8)( 9,11,10,12)(13,16,14,15)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $36$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 9,12)(10,11)(13,16)(14,15)$
$ 12, 3, 3 $ $96$ $12$ $( 1,17, 7)( 2,18, 8)( 3,13,10, 5,16,12, 4,14, 9, 6,15,11)$
$ 6, 6, 6 $ $96$ $6$ $( 1,18, 7, 2,17, 8)( 3,14, 9, 5,16,11)( 4,13,10, 6,15,12)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 3, 6, 4, 5)( 7,18, 8,17)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 2, 2, 2 $ $72$ $4$ $( 1, 2)( 3, 5)( 4, 6)( 7,17, 8,18)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $36$ $2$ $( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 4, 4, 1, 1 $ $144$ $4$ $( 3,17, 6, 8)( 4,18, 5, 7)( 9,15,14,12)(10,16,13,11)$
$ 8, 4, 4, 2 $ $144$ $8$ $( 1, 2)( 3,18, 6, 7, 4,17, 5, 8)( 9,15,13,12)(10,16,14,11)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 3,17, 4,18)( 5, 7)( 6, 8)( 9,14,10,13)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 2)( 3,18)( 4,17)( 5, 8)( 6, 7)( 9,13)(10,14)(11,12)(15,16)$
$ 6, 6, 6 $ $192$ $6$ $( 1,17,15, 9,11, 5)( 2,18,16,10,12, 6)( 3, 7,14, 4, 8,13)$

Group invariants

Order:  $1152=2^{7} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:    
      2  7  6  7  3  3  1  5  5  5  5   2  2  5  4  5  3  3  4  4  1
      3  2  1  .  2  1  2  1  1  .  .   1  1  .  .  .  .  .  .  1  1

        1a 2a 2b 3a 6a 3b 4a 2c 4b 2d 12a 6b 4c 4d 2e 4e 8a 4f 2f 6c
     2P 1a 1a 1a 3a 3a 3b 2a 1a 2a 1a  6a 3a 2b 2a 1a 2e 4c 2b 1a 3b
     3P 1a 2a 2b 1a 2a 1a 4a 2c 4b 2d  4a 2c 4c 4d 2e 4e 8a 4f 2f 2f
     5P 1a 2a 2b 3a 6a 3b 4a 2c 4b 2d 12a 6b 4c 4d 2e 4e 8a 4f 2f 6c
     7P 1a 2a 2b 3a 6a 3b 4a 2c 4b 2d 12a 6b 4c 4d 2e 4e 8a 4f 2f 6c
    11P 1a 2a 2b 3a 6a 3b 4a 2c 4b 2d 12a 6b 4c 4d 2e 4e 8a 4f 2f 6c

X.1      1  1  1  1  1  1  1  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  1  1  1 -1 -1  1  1  1
X.3      1  1  1  1  1  1 -1 -1 -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  1  1  1 -1 -1 -1 -1 -1
X.5      2  2  2  2  2  2  .  .  .  .   .  . -2 -2 -2  .  .  .  .  .
X.6      4  4  4  1  1 -2  2  2  2  2  -1 -1  .  .  .  .  .  .  .  .
X.7      4  4  4  1  1 -2 -2 -2 -2 -2   1  1  .  .  .  .  .  .  .  .
X.8      4  4  4 -2 -2  1  .  .  .  .   .  .  .  .  .  .  . -2 -2  1
X.9      4  4  4 -2 -2  1  .  .  .  .   .  .  .  .  .  .  .  2  2 -1
X.10     6  2 -2  3 -1  . -2 -4  2  .   1 -1 -2  .  2  .  .  .  .  .
X.11     6  2 -2  3 -1  . -4 -2  .  2  -1  1  2  . -2  .  .  .  .  .
X.12     6  2 -2  3 -1  .  2  4 -2  .  -1  1 -2  .  2  .  .  .  .  .
X.13     6  2 -2  3 -1  .  4  2  . -2   1 -1  2  . -2  .  .  .  .  .
X.14     9 -3  1  .  .  . -3  3  1 -1   .  .  1 -1  1 -1  1  1 -3  .
X.15     9 -3  1  .  .  . -3  3  1 -1   .  .  1 -1  1  1 -1 -1  3  .
X.16     9 -3  1  .  .  .  3 -3 -1  1   .  .  1 -1  1 -1  1 -1  3  .
X.17     9 -3  1  .  .  .  3 -3 -1  1   .  .  1 -1  1  1 -1  1 -3  .
X.18    12  4 -4 -3  1  .  2 -2  2 -2  -1  1  .  .  .  .  .  .  .  .
X.19    12  4 -4 -3  1  . -2  2 -2  2   1 -1  .  .  .  .  .  .  .  .
X.20    18 -6  2  .  .  .  .  .  .  .   .  . -2  2 -2  .  .  .  .  .