# Properties

 Label 28T10 Degree $28$ Order $56$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{28}$

# Related objects

## Group action invariants

 Degree $n$: $28$ Transitive number $t$: $10$ Group: $D_{28}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,12,21,3,14,24,6,15,25,8,18,28,9,19,2,11,22,4,13,23,5,16,26,7,17,27,10,20), (1,28)(2,27)(3,25)(4,26)(5,23)(6,24)(7,22)(8,21)(9,20)(10,19)(11,17)(12,18)(13,16)(14,15)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$14$:  $D_{7}$
$28$:  $D_{14}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

## Low degree siblings

28T10

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $14$ $2$ $( 3,27)( 4,28)( 5,25)( 6,26)( 7,24)( 8,23)( 9,22)(10,21)(11,19)(12,20)(13,18) (14,17)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $14$ $2$ $( 1, 3)( 2, 4)( 5,28)( 6,27)( 7,25)( 8,26)( 9,23)(10,24)(11,22)(12,21)(13,19) (14,20)(15,17)(16,18)$ $28$ $2$ $28$ $( 1, 3, 6, 8, 9,11,13,16,17,20,21,24,25,28, 2, 4, 5, 7,10,12,14,15,18,19,22, 23,26,27)$ $28$ $2$ $28$ $( 1, 4, 6, 7, 9,12,13,15,17,19,21,23,25,27, 2, 3, 5, 8,10,11,14,16,18,20,22, 24,26,28)$ $7, 7, 7, 7$ $2$ $7$ $( 1, 5, 9,14,17,22,25)( 2, 6,10,13,18,21,26)( 3, 7,11,15,20,23,28) ( 4, 8,12,16,19,24,27)$ $14, 14$ $2$ $14$ $( 1, 6, 9,13,17,21,25, 2, 5,10,14,18,22,26)( 3, 8,11,16,20,24,28, 4, 7,12,15, 19,23,27)$ $28$ $2$ $28$ $( 1, 7,13,19,25, 3,10,16,22,28, 6,12,17,23, 2, 8,14,20,26, 4, 9,15,21,27, 5, 11,18,24)$ $28$ $2$ $28$ $( 1, 8,13,20,25, 4,10,15,22,27, 6,11,17,24, 2, 7,14,19,26, 3, 9,16,21,28, 5, 12,18,23)$ $7, 7, 7, 7$ $2$ $7$ $( 1, 9,17,25, 5,14,22)( 2,10,18,26, 6,13,21)( 3,11,20,28, 7,15,23) ( 4,12,19,27, 8,16,24)$ $14, 14$ $2$ $14$ $( 1,10,17,26, 5,13,22, 2, 9,18,25, 6,14,21)( 3,12,20,27, 7,16,23, 4,11,19,28, 8,15,24)$ $28$ $2$ $28$ $( 1,11,21, 4,14,23, 6,16,25, 7,18,27, 9,20, 2,12,22, 3,13,24, 5,15,26, 8,17, 28,10,19)$ $28$ $2$ $28$ $( 1,12,21, 3,14,24, 6,15,25, 8,18,28, 9,19, 2,11,22, 4,13,23, 5,16,26, 7,17, 27,10,20)$ $14, 14$ $2$ $14$ $( 1,13,25,10,22, 6,17, 2,14,26, 9,21, 5,18)( 3,16,28,12,23, 8,20, 4,15,27,11, 24, 7,19)$ $7, 7, 7, 7$ $2$ $7$ $( 1,14,25, 9,22, 5,17)( 2,13,26,10,21, 6,18)( 3,15,28,11,23, 7,20) ( 4,16,27,12,24, 8,19)$ $4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,27,14,28)$

## Group invariants

 Order: $56=2^{3} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [56, 5]
 Character table:  2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 7 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 28a 28b 7a 14a 28c 28d 7b 14b 28e 28f 14c 7c 4a 2P 1a 1a 1a 1a 14a 14a 7b 7b 14c 14c 7c 7c 14b 14b 7a 7a 2b 3P 1a 2a 2b 2c 28d 28c 7c 14c 28e 28f 7a 14a 28b 28a 14b 7b 4a 5P 1a 2a 2b 2c 28e 28f 7b 14b 28a 28b 7c 14c 28d 28c 14a 7a 4a 7P 1a 2a 2b 2c 4a 4a 1a 2b 4a 4a 1a 2b 4a 4a 2b 1a 4a 11P 1a 2a 2b 2c 28c 28d 7c 14c 28f 28e 7a 14a 28a 28b 14b 7b 4a 13P 1a 2a 2b 2c 28b 28a 7a 14a 28d 28c 7b 14b 28f 28e 14c 7c 4a 17P 1a 2a 2b 2c 28c 28d 7c 14c 28f 28e 7a 14a 28a 28b 14b 7b 4a 19P 1a 2a 2b 2c 28f 28e 7b 14b 28b 28a 7c 14c 28c 28d 14a 7a 4a 23P 1a 2a 2b 2c 28e 28f 7b 14b 28a 28b 7c 14c 28d 28c 14a 7a 4a X.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 X.3 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 X.5 2 . -2 . . . 2 -2 . . 2 -2 . . -2 2 . X.6 2 . 2 . A A -B -B C C -C -C B B -A -A -2 X.7 2 . 2 . B B -C -C A A -A -A C C -B -B -2 X.8 2 . 2 . C C -A -A B B -B -B A A -C -C -2 X.9 2 . 2 . -A -A -B -B -C -C -C -C -B -B -A -A 2 X.10 2 . 2 . -B -B -C -C -A -A -A -A -C -C -B -B 2 X.11 2 . 2 . -C -C -A -A -B -B -B -B -A -A -C -C 2 X.12 2 . -2 . D -D -C C -F F -A A E -E B -B . X.13 2 . -2 . E -E -A A D -D -B B F -F C -C . X.14 2 . -2 . F -F -B B E -E -C C -D D A -A . X.15 2 . -2 . -F F -B B -E E -C C D -D A -A . X.16 2 . -2 . -E E -A A -D D -B B -F F C -C . X.17 2 . -2 . -D D -C C F -F -A A -E E B -B . A = -E(7)^3-E(7)^4 B = -E(7)-E(7)^6 C = -E(7)^2-E(7)^5 D = -E(28)^3+E(28)^11 E = -E(28)^15+E(28)^27 F = -E(28)^19+E(28)^23