Galois Group: 28T15

• Label: 28T15
• Order: \(84\)
• n: \(28\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_2\times F_7$

Group action invariants

Degree $n$ :		$28$
Transitive number $t$ :		$15$
Group :		$C_2\times F_7$
Parity:		$1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2)(3,8,19,27,23,12)(4,7,20,28,24,11)(5,14,9,26,17,22)(6,13,10,25,18,21)(15,16), (1,12,18,16,26,4)(2,11,17,15,25,3)(5,19)(6,20)(7,9,28,21,23,13)(8,10,27,22,24,14)
$|\Aut(F/K)|$:		$4$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$ x 3
3:	$C_3$
4:	$C_2^2$
6:	$C_6$ x 3
12:	$C_6\times C_2$
42:	$F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $F_7$

Degree 14: $F_7$, $F_7 \times C_2$ x 2

Low degree siblings

14T7 x 2, 42T10 x 2

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

Group invariants

Order:		$84=2^{2} \cdot 3 \cdot 7$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[84, 7]

Character table:

2 2 2 2 2 2 2 2 2 2 1 2 2 1 2 3 1 1 1 1 1 1 1 1 1 . 1 1 . 1 7 1 . . . . . . . . 1 . . 1 1 1a 3a 3b 6a 6b 2a 2b 6c 6d 14a 6e 6f 7a 2c 2P 1a 3b 3a 3a 3b 1a 1a 3a 3b 7a 3b 3a 7a 1a 3P 1a 1a 1a 2a 2a 2a 2b 2b 2b 14a 2c 2c 7a 2c 5P 1a 3b 3a 6b 6a 2a 2b 6d 6c 14a 6f 6e 7a 2c 7P 1a 3a 3b 6a 6b 2a 2b 6c 6d 2c 6e 6f 1a 2c 11P 1a 3b 3a 6b 6a 2a 2b 6d 6c 14a 6f 6e 7a 2c 13P 1a 3a 3b 6a 6b 2a 2b 6c 6d 14a 6e 6f 7a 2c 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 A /A -/A -A -1 -1 -/A -A 1 A /A 1 1 X.6 1 /A A -A -/A -1 -1 -A -/A 1 /A A 1 1 X.7 1 A /A -/A -A -1 1 /A A -1 -A -/A 1 -1 X.8 1 /A A -A -/A -1 1 A /A -1 -/A -A 1 -1 X.9 1 A /A /A A 1 -1 -/A -A -1 -A -/A 1 -1 X.10 1 /A A A /A 1 -1 -A -/A -1 -/A -A 1 -1 X.11 1 A /A /A A 1 1 /A A 1 A /A 1 1 X.12 1 /A A A /A 1 1 A /A 1 /A A 1 1 X.13 6 . . . . . . . . 1 . . -1 -6 X.14 6 . . . . . . . . -1 . . -1 6 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3