Properties

 Label 16T22 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_{16} : C_2$

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Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $22$ Group : $C_{16} : C_2$ Parity: $-1$ Primitive: No Nilpotency class: $2$ Generators: (1,12,7,10,5,16,3,14,2,11,8,9,6,15,4,13), (1,12,8,9,5,16,4,13,2,11,7,10,6,15,3,14) $|\Aut(F/K)|$: $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_8$ x 2, $C_4\times C_2$
16:  $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

32T8

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $8, 8$ $2$ $8$ $( 1, 3, 6, 7, 2, 4, 5, 8)( 9,11,14,16,10,12,13,15)$ $8, 8$ $1$ $8$ $( 1, 3, 6, 7, 2, 4, 5, 8)( 9,12,14,15,10,11,13,16)$ $8, 8$ $1$ $8$ $( 1, 4, 6, 8, 2, 3, 5, 7)( 9,11,14,16,10,12,13,15)$ $4, 4, 4, 4$ $1$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$ $4, 4, 4, 4$ $1$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$ $8, 8$ $1$ $8$ $( 1, 7, 5, 3, 2, 8, 6, 4)( 9,15,13,12,10,16,14,11)$ $8, 8$ $2$ $8$ $( 1, 7, 5, 3, 2, 8, 6, 4)( 9,16,13,11,10,15,14,12)$ $8, 8$ $1$ $8$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,16,13,11,10,15,14,12)$ $16$ $2$ $16$ $( 1, 9, 3,12, 6,14, 7,15, 2,10, 4,11, 5,13, 8,16)$ $16$ $2$ $16$ $( 1, 9, 4,11, 6,14, 8,16, 2,10, 3,12, 5,13, 7,15)$ $16$ $2$ $16$ $( 1,11, 7, 9, 5,15, 3,13, 2,12, 8,10, 6,16, 4,14)$ $16$ $2$ $16$ $( 1,11, 8,10, 5,15, 4,14, 2,12, 7, 9, 6,16, 3,13)$ $16$ $2$ $16$ $( 1,13, 3,16, 6, 9, 7,12, 2,14, 4,15, 5,10, 8,11)$ $16$ $2$ $16$ $( 1,13, 4,15, 6, 9, 8,11, 2,14, 3,16, 5,10, 7,12)$ $16$ $2$ $16$ $( 1,15, 7,13, 5,12, 3,10, 2,16, 8,14, 6,11, 4, 9)$ $16$ $2$ $16$ $( 1,15, 8,14, 5,12, 4, 9, 2,16, 7,13, 6,11, 3,10)$

Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 17]
 Character table:  2 5 4 5 4 5 5 5 4 5 5 4 5 4 4 4 4 4 4 4 1a 2a 2b 8a 8b 8c 4a 4b 4c 8d 8e 8f 16a 16b 16c 16d 16e 16f 16g 2P 1a 1a 1a 4c 4c 4c 2b 2b 2b 4a 4a 4a 8b 8c 8d 8f 8b 8c 8d 3P 1a 2a 2b 8e 8d 8f 4c 4b 4a 8b 8a 8c 16c 16d 16a 16b 16g 16h 16e 5P 1a 2a 2b 8a 8c 8b 4a 4b 4c 8f 8e 8d 16f 16e 16h 16g 16b 16a 16d 7P 1a 2a 2b 8e 8f 8d 4c 4b 4a 8c 8a 8b 16h 16g 16f 16e 16d 16c 16b 11P 1a 2a 2b 8e 8d 8f 4c 4b 4a 8b 8a 8c 16c 16d 16a 16b 16g 16h 16e 13P 1a 2a 2b 8a 8c 8b 4a 4b 4c 8f 8e 8d 16f 16e 16h 16g 16b 16a 16d X.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 X.3 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 X.5 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 A -A -A A -A A A X.6 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -A A A -A A -A -A X.7 1 -1 1 A -A -A -1 1 -1 A -A A D -D -/D /D D -D -/D X.8 1 -1 1 A -A -A -1 1 -1 A -A A -D D /D -/D -D D /D X.9 1 -1 1 -A A A -1 1 -1 -A A -A -/D /D D -D -/D /D D X.10 1 -1 1 -A A A -1 1 -1 -A A -A /D -/D -D D /D -/D -D X.11 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 A A -A -A A A -A X.12 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -A -A A A -A -A A X.13 1 1 1 A A A -1 -1 -1 -A -A -A -/D -/D D D /D /D -D X.14 1 1 1 A A A -1 -1 -1 -A -A -A /D /D -D -D -/D -/D D X.15 1 1 1 -A -A -A -1 -1 -1 A A A D D -/D -/D -D -D /D X.16 1 1 1 -A -A -A -1 -1 -1 A A A -D -D /D /D D D -/D X.17 2 . -2 . B -B C . -C -/B . /B . . . . . . . X.18 2 . -2 . -/B /B -C . C B . -B . . . . . . . X.19 2 . -2 . /B -/B -C . C -B . B . . . . . . . X.20 2 . -2 . -B B C . -C /B . -/B . . . . . . . 2 4 16h 2P 8f 3P 16f 5P 16c 7P 16a 11P 16f 13P 16c X.1 1 X.2 -1 X.3 1 X.4 -1 X.5 -A X.6 A X.7 /D X.8 -/D X.9 -D X.10 D X.11 -A X.12 A X.13 -D X.14 D X.15 /D X.16 -/D X.17 . X.18 . X.19 . X.20 . A = -E(4) = -Sqrt(-1) = -i B = -2*E(8) C = -2*E(4) = -2*Sqrt(-1) = -2i D = -E(8)