# Properties

 Label 28T3 Order $$28$$ n $$28$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_7:C_4$

## Group action invariants

 Degree $n$ : $28$ Transitive number $t$ : $3$ Group : $C_7:C_4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3,2,4)(5,27,6,28)(7,25,8,26)(9,24,10,23)(11,22,12,21)(13,19,14,20)(15,18,16,17), (1,8,2,7)(3,5,4,6)(9,28,10,27)(11,26,12,25)(13,23,14,24)(15,22,16,21)(17,19,18,20) $|\Aut(F/K)|$: $28$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
14:  $D_{7}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $28=2^{2} \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [28, 1]
 Character table:  2 2 2 2 2 1 1 1 1 1 1 7 1 1 . . 1 1 1 1 1 1 1a 2a 4a 4b 7a 14a 7b 14b 7c 14c 2P 1a 1a 2a 2a 7b 7b 7c 7c 7a 7a 3P 1a 2a 4b 4a 7c 14c 7a 14a 7b 14b 5P 1a 2a 4a 4b 7b 14b 7c 14c 7a 14a 7P 1a 2a 4b 4a 1a 2a 1a 2a 1a 2a 11P 1a 2a 4b 4a 7c 14c 7a 14a 7b 14b 13P 1a 2a 4a 4b 7a 14a 7b 14b 7c 14c X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 1 X.3 1 -1 A -A 1 -1 1 -1 1 -1 X.4 1 -1 -A A 1 -1 1 -1 1 -1 X.5 2 -2 . . B -B D -D C -C X.6 2 -2 . . C -C B -B D -D X.7 2 -2 . . D -D C -C B -B X.8 2 2 . . B B D D C C X.9 2 2 . . C C B B D D X.10 2 2 . . D D C C B B A = -E(4) = -Sqrt(-1) = -i B = E(7)^3+E(7)^4 C = E(7)^2+E(7)^5 D = E(7)+E(7)^6