Galois group: 28T15

• Label: 28T15
• Degree: $28$
• Order: $84$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2\times F_7$

Group invariants

Abstract group:		$C_2\times F_7$
Order:		$84=2^{2} \cdot 3 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$28$
Transitive number $t$:		$15$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,2)(3,8,19,27,23,12)(4,7,20,28,24,11)(5,14,9,26,17,22)(6,13,10,25,18,21)(15,16)$, $(1,12,18,16,26,4)(2,11,17,15,25,3)(5,19)(6,20)(7,9,28,21,23,13)(8,10,27,22,24,14)$

Low degree resolvents

$\card{(G/N)}$	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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{28}$	$1$	$1$	$0$	$()$
2A	$2^{14}$	$1$	$2$	$14$	$( 1,16)( 2,15)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26)(13,28)(14,27)$
2B	$2^{14}$	$7$	$2$	$14$	$( 1, 2)( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)$
2C	$2^{14}$	$7$	$2$	$14$	$( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,27)(14,28)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
3A1	$3^{8},1^{4}$	$7$	$3$	$16$	$( 3,19,23)( 4,20,24)( 5, 9,17)( 6,10,18)( 7,28,11)( 8,27,12)(13,25,21)(14,26,22)$
3A-1	$3^{8},1^{4}$	$7$	$3$	$16$	$( 3,23,19)( 4,24,20)( 5,17, 9)( 6,18,10)( 7,11,28)( 8,12,27)(13,21,25)(14,22,26)$
6A1	$6^{4},2^{2}$	$7$	$6$	$22$	$( 1, 2)( 3, 8,19,27,23,12)( 4, 7,20,28,24,11)( 5,14, 9,26,17,22)( 6,13,10,25,18,21)(15,16)$
6A-1	$6^{4},2^{2}$	$7$	$6$	$22$	$( 1, 2)( 3,12,23,27,19, 8)( 4,11,24,28,20, 7)( 5,22,17,26, 9,14)( 6,21,18,25,10,13)(15,16)$
6B1	$6^{4},2^{2}$	$7$	$6$	$22$	$( 1,20,14,16, 6,27)( 2,19,13,15, 5,28)( 3, 9, 7,17,23,21)( 4,10, 8,18,24,22)(11,25)(12,26)$
6B-1	$6^{4},2^{2}$	$7$	$6$	$22$	$( 1,20,22,16, 6, 8)( 2,19,21,15, 5, 7)( 3,13,11,17,28,25)( 4,14,12,18,27,26)( 9,23)(10,24)$
6C1	$6^{4},2^{2}$	$7$	$6$	$22$	$( 1,19,18,11,22,23)( 2,20,17,12,21,24)( 3,26, 7,10,15, 6)( 4,25, 8, 9,16, 5)(13,27)(14,28)$
6C-1	$6^{4},2^{2}$	$7$	$6$	$22$	$( 1,19,26,28,10, 3)( 2,20,25,27, 9, 4)( 5,12,13,24,17,16)( 6,11,14,23,18,15)( 7,22)( 8,21)$
7A	$7^{4}$	$6$	$7$	$24$	$( 1,10,18,26, 6,14,22)( 2, 9,17,25, 5,13,21)( 3,11,19,28, 7,15,23)( 4,12,20,27, 8,16,24)$
14A	$14^{2}$	$6$	$14$	$26$	$( 1,20,10,27,18, 8,26,16, 6,24,14, 4,22,12)( 2,19, 9,28,17, 7,25,15, 5,23,13, 3,21,11)$

Malle's constant $a(G)$:     $1/14$

Character table

		1A	2A	2B	2C	3A1	3A-1	6A1	6A-1	6B1	6B-1	6C1	6C-1	7A	14A
	Size	1	1	7	7	7	7	7	7	7	7	7	7	6	6
	2 P	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1	3A-1	3A1	3A1	3A-1	7A	7A
	3 P	1A	2A	2B	2C	1A	1A	2B	2B	2A	2A	2C	2C	7A	14A
	7 P	1A	2A	2B	2C	3A1	3A-1	6A1	6A-1	6B1	6B-1	6C1	6C-1	1A	2A
	Type
84.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
84.7.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
84.7.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
84.7.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$
84.7.1e1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
84.7.1e2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
84.7.1f1	C	$1$	$-1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$-1$
84.7.1f2	C	$1$	$-1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$-1$
84.7.1g1	C	$1$	$-1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$-1$
84.7.1g2	C	$1$	$-1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$-1$
84.7.1h1	C	$1$	$1$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$
84.7.1h2	C	$1$	$1$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$
84.7.6a	R	$6$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
84.7.6b	R	$6$	$-6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$

Regular extensions

Data not computed