Group action invariants
Degree $n$: | $32$ | |
Transitive number $t$: | $28$ | |
Group: | $C_8.C_4$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $3$ | |
$|\Aut(F/K)|$: | $32$ | |
Generators: | (1,23,12,30,3,21,10,32)(2,24,11,29,4,22,9,31)(5,19,13,28,8,17,15,25)(6,20,14,27,7,18,16,26), (1,18,10,27,3,20,12,26)(2,17,9,28,4,19,11,25)(5,32,15,21,8,30,13,23)(6,31,16,22,7,29,14,24) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$, $C_4\times C_2$, $Q_8$ $16$: $C_4:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2
Degree 8: $C_4\times C_2$, $D_4$, $Q_8$
Low degree siblings
16T49Siblings 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, 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 5,11,14, 3, 8, 9,16)( 2, 6,12,13, 4, 7,10,15)(17,21,27,29,19,23,26,31) (18,22,28,30,20,24,25,32)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 6,11,13, 3, 7, 9,15)( 2, 5,12,14, 4, 8,10,16)(17,24,27,32,19,22,26,30) (18,23,28,31,20,21,25,29)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 7,11,15, 3, 6, 9,13)( 2, 8,12,16, 4, 5,10,14)(17,22,27,30,19,24,26,32) (18,21,28,29,20,23,25,31)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 8,11,16, 3, 5, 9,14)( 2, 7,12,15, 4, 6,10,13)(17,23,27,31,19,21,26,29) (18,24,28,32,20,22,25,30)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 3,11)( 2,10, 4,12)( 5,16, 8,14)( 6,15, 7,13)(17,26,19,27)(18,25,20,28) (21,31,23,29)(22,32,24,30)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,10, 3,12)( 2, 9, 4,11)( 5,15, 8,13)( 6,16, 7,14)(17,28,19,25)(18,27,20,26) (21,30,23,32)(22,29,24,31)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,12, 3,10)( 2,11, 4, 9)( 5,13, 8,15)( 6,14, 7,16)(17,25,19,28)(18,26,20,27) (21,32,23,30)(22,31,24,29)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,17,12,25, 3,19,10,28)( 2,18,11,26, 4,20, 9,27)( 5,31,13,24, 8,29,15,22) ( 6,32,14,23, 7,30,16,21)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,18,10,27, 3,20,12,26)( 2,17, 9,28, 4,19,11,25)( 5,32,15,21, 8,30,13,23) ( 6,31,16,22, 7,29,14,24)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,21,12,32, 3,23,10,30)( 2,22,11,31, 4,24, 9,29)( 5,17,13,25, 8,19,15,28) ( 6,18,14,26, 7,20,16,27)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,22,10,29, 3,24,12,31)( 2,21, 9,30, 4,23,11,32)( 5,18,15,27, 8,20,13,26) ( 6,17,16,28, 7,19,14,25)$ |
Group invariants
Order: | $32=2^{5}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [32, 15] |
Character table: |
2 5 4 5 4 4 4 4 4 5 5 3 3 3 3 1a 2a 2b 8a 8b 8c 8d 4a 4b 4c 8e 8f 8g 8h 2P 1a 1a 1a 4a 4a 4a 4a 2b 2b 2b 4c 4b 4c 4b 3P 1a 2a 2b 8d 8b 8c 8a 4a 4c 4b 8f 8e 8h 8g 5P 1a 2a 2b 8d 8c 8b 8a 4a 4b 4c 8e 8f 8g 8h 7P 1a 2a 2b 8a 8c 8b 8d 4a 4c 4b 8f 8e 8h 8g X.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 X.3 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 X.5 1 -1 1 -1 1 1 -1 1 -1 -1 D -D -D D X.6 1 -1 1 -1 1 1 -1 1 -1 -1 -D D D -D X.7 1 -1 1 1 -1 -1 1 1 -1 -1 D -D D -D X.8 1 -1 1 1 -1 -1 1 1 -1 -1 -D D -D D X.9 2 2 2 . . . . -2 -2 -2 . . . . X.10 2 -2 2 . . . . -2 2 2 . . . . X.11 2 . -2 A B -B -A . C -C . . . . X.12 2 . -2 A -B B -A . -C C . . . . X.13 2 . -2 -A B -B A . -C C . . . . X.14 2 . -2 -A -B B A . C -C . . . . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2 B = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 C = -2*E(4) = -2*Sqrt(-1) = -2i D = -E(4) = -Sqrt(-1) = -i |