# Properties

 Label 16T24 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2^2 : C_8$

# Related objects

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $24$ Group : $C_2^2 : C_8$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,3,5,7,10,11,14,15)(2,4,6,8,9,12,13,16), (1,10)(2,9)(3,4)(5,14)(6,13)(7,8)(11,12)(15,16) $|\Aut(F/K)|$: $8$

## 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:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_8$, $C_8:C_2$, $C_2^2:C_4$

## Low degree siblings

16T24, 32T10

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

## Group invariants

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