Galois group: 21T41

• Label: 21T41
• Degree: $21$
• Order: $6174$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_7^3:(C_3\times S_3)$

Group invariants

Abstract group:		$C_7^3:(C_3\times S_3)$
Order:		$6174=2 \cdot 3^{2} \cdot 7^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$41$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,8,4,11,2,9)(3,10,5,12,6,13)(7,14)(15,18,19,17,21,20)$, $(1,10,18,2,9,16,3,8,21,4,14,19,5,13,17,6,12,15,7,11,20)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$18$:	$S_3\times C_3$
$42$:	$F_7$
$126$:	21T10
$882$:	14T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

21T41 x 5, 42T465 x 6, 42T472 x 3, 42T475 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	$1^{21}$	$1$	$1$	$0$	$()$
2A	$2^{10},1$	$147$	$2$	$10$	$( 2, 7)( 3, 6)( 4, 5)( 8,20)( 9,19)(10,18)(11,17)(12,16)(13,15)(14,21)$
3A	$3^{7}$	$98$	$3$	$14$	$( 1,12,16)( 2, 8,19)( 3,11,15)( 4,14,18)( 5,10,21)( 6,13,17)( 7, 9,20)$
3B1	$3^{7}$	$98$	$3$	$14$	$( 1,17,14)( 2,15,13)( 3,20,12)( 4,18,11)( 5,16,10)( 6,21, 9)( 7,19, 8)$
3B-1	$3^{7}$	$98$	$3$	$14$	$( 1,14,17)( 2,13,15)( 3,12,20)( 4,11,18)( 5,10,16)( 6, 9,21)( 7, 8,19)$
3C1	$3^{6},1^{3}$	$343$	$3$	$12$	$( 1, 5, 7)( 3, 6, 4)( 8,12,14)(10,13,11)(15,20,19)(16,17,21)$
3C-1	$3^{6},1^{3}$	$343$	$3$	$12$	$( 1, 7, 5)( 3, 4, 6)( 8,14,12)(10,11,13)(15,19,20)(16,21,17)$
6A1	$6^{3},2,1$	$1029$	$6$	$16$	$( 1, 4, 5, 3, 7, 6)( 8,20,12,19,14,15)( 9,18)(10,16,13,17,11,21)$
6A-1	$6^{3},2,1$	$1029$	$6$	$16$	$( 1, 6, 7, 3, 5, 4)( 8,15,14,19,12,20)( 9,18)(10,21,11,17,13,16)$
7A	$7^{3}$	$6$	$7$	$18$	$( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
7B1	$7^{3}$	$6$	$7$	$18$	$( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$
7B-1	$7^{3}$	$6$	$7$	$18$	$( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
7C1	$7^{2},1^{7}$	$9$	$7$	$12$	$( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7C-1	$7^{2},1^{7}$	$9$	$7$	$12$	$( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
7D	$7^{3}$	$18$	$7$	$18$	$( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
7E	$7^{3}$	$18$	$7$	$18$	$( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
7F	$7,1^{14}$	$18$	$7$	$6$	$( 8,13,11, 9,14,12,10)$
7G	$7^{3}$	$18$	$7$	$18$	$( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7H	$7^{3}$	$18$	$7$	$18$	$( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$
7I	$7^{3}$	$18$	$7$	$18$	$( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
7J	$7^{2},1^{7}$	$18$	$7$	$12$	$( 1, 3, 5, 7, 2, 4, 6)(15,21,20,19,18,17,16)$
7K1	$7^{2},1^{7}$	$18$	$7$	$12$	$( 1, 6, 4, 2, 7, 5, 3)(15,19,16,20,17,21,18)$
7K-1	$7^{2},1^{7}$	$18$	$7$	$12$	$( 1, 2, 3, 4, 5, 6, 7)(15,20,18,16,21,19,17)$
7L1	$7^{3}$	$18$	$7$	$18$	$( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
7L-1	$7^{3}$	$18$	$7$	$18$	$( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
7M1	$7^{3}$	$18$	$7$	$18$	$( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
7M-1	$7^{3}$	$18$	$7$	$18$	$( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7N1	$7^{2},1^{7}$	$18$	$7$	$12$	$( 1, 2, 3, 4, 5, 6, 7)(15,17,19,21,16,18,20)$
7N-1	$7^{2},1^{7}$	$18$	$7$	$12$	$( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
7O1	$7^{3}$	$18$	$7$	$18$	$( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
7O-1	$7^{3}$	$18$	$7$	$18$	$( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
14A1	$14,2^{3},1$	$441$	$14$	$16$	$( 2, 7)( 3, 6)( 4, 5)( 8,16, 9,15,10,21,11,20,12,19,13,18,14,17)$
14A-1	$14,2^{3},1$	$441$	$14$	$16$	$( 2, 7)( 3, 6)( 4, 5)( 8,17,14,18,13,19,12,20,11,21,10,15, 9,16)$
21A1	$21$	$294$	$21$	$20$	$( 1,13,18, 2, 9,21, 3,12,17, 4, 8,20, 5,11,16, 6,14,19, 7,10,15)$
21A2	$21$	$294$	$21$	$20$	$( 1,18, 9, 3,17, 8, 5,16,14, 7,15,13, 2,21,12, 4,20,11, 6,19,10)$
21B1	$21$	$294$	$21$	$20$	$( 1,21,11, 2,19,10, 3,17, 9, 4,15, 8, 5,20,14, 6,18,13, 7,16,12)$
21B-1	$21$	$294$	$21$	$20$	$( 1,12,16, 7,13,18, 6,14,20, 5, 8,15, 4, 9,17, 3,10,19, 2,11,21)$
21B2	$21$	$294$	$21$	$20$	$( 1,11,19, 3, 9,15, 5,14,18, 7,12,21, 2,10,17, 4, 8,20, 6,13,16)$
21B-2	$21$	$294$	$21$	$20$	$( 1,16,13, 6,20, 8, 4,17,10, 2,21,12, 7,18,14, 5,15, 9, 3,19,11)$

Malle's constant $a(G)$:     $1/6$

Character table

39 x 39 character table

Regular extensions

Data not computed