# Properties

 Label 24T33 Degree $24$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_6:C_4$

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## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $33$ Group: $D_6:C_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,15,6,19,9,24,13,3,17,7,21,11)(2,16,5,20,10,23,14,4,18,8,22,12)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$12$:  $D_{6}$
$16$:  $C_2^2:C_4$
$24$:  $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

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

Degree 6: $D_{6}$

Degree 8: $C_2^2:C_4$

Degree 12: $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

## Low degree siblings

24T33

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $6$ $2$ $( 3,12)( 4,11)( 5,22)( 6,21)( 7, 8)( 9,17)(10,18)(15,23)(16,24)(19,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$ $12, 12$ $2$ $12$ $( 1, 3, 6, 7, 9,11,13,15,17,19,21,24)( 2, 4, 5, 8,10,12,14,16,18,20,22,23)$ $4, 4, 4, 4, 4, 4$ $6$ $4$ $( 1, 3,14,16)( 2, 4,13,15)( 5,23,17,11)( 6,24,18,12)( 7,10,20,21)( 8, 9,19,22)$ $12, 12$ $2$ $12$ $( 1, 4, 6, 8, 9,12,13,16,17,20,21,23)( 2, 3, 5, 7,10,11,14,15,18,19,22,24)$ $6, 6, 6, 6$ $2$ $6$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,22)(10,21)(13,18)(14,17)(19,24) (20,23)$ $6, 6, 6, 6$ $2$ $6$ $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$ $4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 7,13,19)( 2, 8,14,20)( 3, 9,15,21)( 4,10,16,22)( 5,12,18,23)( 6,11,17,24)$ $4, 4, 4, 4, 4, 4$ $6$ $4$ $( 1, 7,14,20)( 2, 8,13,19)( 3,18,16, 6)( 4,17,15, 5)( 9,24,22,12)(10,23,21,11)$ $3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$ $6, 6, 6, 6$ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,24) (12,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$ $12, 12$ $2$ $12$ $( 1,15, 6,19, 9,24,13, 3,17, 7,21,11)( 2,16, 5,20,10,23,14, 4,18, 8,22,12)$ $12, 12$ $2$ $12$ $( 1,16, 6,20, 9,23,13, 4,17, 8,21,12)( 2,15, 5,19,10,24,14, 3,18, 7,22,11)$ $4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,19,13, 7)( 2,20,14, 8)( 3,21,15, 9)( 4,22,16,10)( 5,23,18,12)( 6,24,17,11)$

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [48, 14]
 Character table:  2 4 3 4 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 1 . 1 1 . 1 1 . 1 1 . 1 1 1 1 1 1 1 1a 2a 2b 12a 4a 12b 6a 2c 6b 4b 4c 3a 6c 2d 2e 12c 12d 4d 2P 1a 1a 1a 6b 2e 6b 3a 1a 3a 2d 2e 3a 3a 1a 1a 6b 6b 2d 3P 1a 2a 2b 4b 4c 4b 2e 2c 2d 4d 4a 1a 2b 2d 2e 4d 4d 4b 5P 1a 2a 2b 12b 4a 12a 6a 2c 6b 4b 4c 3a 6c 2d 2e 12d 12c 4d 7P 1a 2a 2b 12c 4c 12d 6a 2c 6b 4d 4a 3a 6c 2d 2e 12a 12b 4b 11P 1a 2a 2b 12d 4c 12c 6a 2c 6b 4d 4a 3a 6c 2d 2e 12b 12a 4b X.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 X.3 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 X.5 1 -1 1 A -A A -1 1 -1 -A A 1 1 -1 -1 -A -A A X.6 1 -1 1 -A A -A -1 1 -1 A -A 1 1 -1 -1 A A -A X.7 1 1 1 A A A -1 -1 -1 -A -A 1 1 -1 -1 -A -A A X.8 1 1 1 -A -A -A -1 -1 -1 A A 1 1 -1 -1 A A -A X.9 2 . -2 . . . -2 . 2 . . 2 -2 2 -2 . . . X.10 2 . -2 . . . 2 . -2 . . 2 -2 -2 2 . . . X.11 2 . 2 -1 . -1 -1 . -1 2 . -1 -1 2 2 -1 -1 2 X.12 2 . 2 1 . 1 -1 . -1 -2 . -1 -1 2 2 1 1 -2 X.13 2 . -2 B . -B 1 . -1 . . -1 1 2 -2 B -B . X.14 2 . -2 -B . B 1 . -1 . . -1 1 2 -2 -B B . X.15 2 . -2 C . -C -1 . 1 . . -1 1 -2 2 -C C . X.16 2 . -2 -C . C -1 . 1 . . -1 1 -2 2 C -C . X.17 2 . 2 A . A 1 . 1 D . -1 -1 -2 -2 -A -A -D X.18 2 . 2 -A . -A 1 . 1 -D . -1 -1 -2 -2 A A D A = -E(4) = -Sqrt(-1) = -i B = -E(3)+E(3)^2 = -Sqrt(-3) = -i3 C = -E(12)^7+E(12)^11 = Sqrt(3) = r3 D = -2*E(4) = -2*Sqrt(-1) = -2i