Galois group: 42T47

• Label: 42T47
• Degree: $42$
• Order: $252$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_3^2:C_{28}$

Group invariants

Abstract group:		$C_3^2:C_{28}$
Order:		$252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$42$
Transitive number $t$:		$47$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$7$
Generators:		$(1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41)$, $(1,17,31,6,19,35,7,24,38,10,27,40,14,28,2,16,32,5,21,36,8,22,37,12,25,42,13,30)(3,18,33,4,20,34,9,23,39,11,26,41,15,29)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$7$:	$C_7$
$14$:	$C_{14}$
$28$:	$C_{28}$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 7: $C_7$

Degree 14: $C_{14}$

Degree 21: None

Low degree siblings

42T47

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	$1^{42}$	$1$	$1$	$0$	$()$
2A	$2^{14},1^{14}$	$9$	$2$	$14$	$( 1, 2)( 5, 6)( 7, 8)(10,12)(13,14)(16,17)(19,21)(22,24)(25,27)(28,30)(31,32)(35,36)(37,38)(40,42)$
3A	$3^{7},1^{21}$	$4$	$3$	$14$	$( 4, 5, 6)(10,12,11)(16,18,17)(22,24,23)(28,30,29)(34,35,36)(40,41,42)$
3B	$3^{14}$	$4$	$3$	$28$	$( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41)$
4A1	$4^{7},2^{7}$	$9$	$4$	$28$	$( 1,24, 2,22)( 3,23)( 4,26)( 5,25, 6,27)( 7,28, 8,30)( 9,29)(10,32,12,31)(11,33)(13,35,14,36)(15,34)(16,37,17,38)(18,39)(19,40,21,42)(20,41)$
4A-1	$4^{7},2^{7}$	$9$	$4$	$28$	$( 1,22, 2,24)( 3,23)( 4,26)( 5,27, 6,25)( 7,30, 8,28)( 9,29)(10,31,12,32)(11,33)(13,36,14,35)(15,34)(16,38,17,37)(18,39)(19,42,21,40)(20,41)$
7A1	$7^{6}$	$1$	$7$	$36$	$( 1,19,38,14,32, 8,25)( 2,21,37,13,31, 7,27)( 3,20,39,15,33, 9,26)( 4,23,41,18,34,11,29)( 5,22,42,17,35,10,28)( 6,24,40,16,36,12,30)$
7A-1	$7^{6}$	$1$	$7$	$36$	$( 1,25, 8,32,14,38,19)( 2,27, 7,31,13,37,21)( 3,26, 9,33,15,39,20)( 4,29,11,34,18,41,23)( 5,28,10,35,17,42,22)( 6,30,12,36,16,40,24)$
7A2	$7^{6}$	$1$	$7$	$36$	$( 1,38,32,25,19,14, 8)( 2,37,31,27,21,13, 7)( 3,39,33,26,20,15, 9)( 4,41,34,29,23,18,11)( 5,42,35,28,22,17,10)( 6,40,36,30,24,16,12)$
7A-2	$7^{6}$	$1$	$7$	$36$	$( 1, 8,14,19,25,32,38)( 2, 7,13,21,27,31,37)( 3, 9,15,20,26,33,39)( 4,11,18,23,29,34,41)( 5,10,17,22,28,35,42)( 6,12,16,24,30,36,40)$
7A3	$7^{6}$	$1$	$7$	$36$	$( 1,14,25,38, 8,19,32)( 2,13,27,37, 7,21,31)( 3,15,26,39, 9,20,33)( 4,18,29,41,11,23,34)( 5,17,28,42,10,22,35)( 6,16,30,40,12,24,36)$
7A-3	$7^{6}$	$1$	$7$	$36$	$( 1,32,19, 8,38,25,14)( 2,31,21, 7,37,27,13)( 3,33,20, 9,39,26,15)( 4,34,23,11,41,29,18)( 5,35,22,10,42,28,17)( 6,36,24,12,40,30,16)$
14A1	$14^{2},7^{2}$	$9$	$14$	$38$	$( 1,37,32,27,19,13, 8, 2,38,31,25,21,14, 7)( 3,39,33,26,20,15, 9)( 4,41,34,29,23,18,11)( 5,40,35,30,22,16,10, 6,42,36,28,24,17,12)$
14A-1	$14^{2},7^{2}$	$9$	$14$	$38$	$( 1, 7,14,21,25,31,38, 2, 8,13,19,27,32,37)( 3, 9,15,20,26,33,39)( 4,11,18,23,29,34,41)( 5,12,17,24,28,36,42, 6,10,16,22,30,35,40)$
14A3	$14^{2},7^{2}$	$9$	$14$	$38$	$( 1,27, 8,31,14,37,19, 2,25, 7,32,13,38,21)( 3,26, 9,33,15,39,20)( 4,29,11,34,18,41,23)( 5,30,10,36,17,40,22, 6,28,12,35,16,42,24)$
14A-3	$14^{2},7^{2}$	$9$	$14$	$38$	$( 1,21,38,13,32, 7,25, 2,19,37,14,31, 8,27)( 3,20,39,15,33, 9,26)( 4,23,41,18,34,11,29)( 5,24,42,16,35,12,28, 6,22,40,17,36,10,30)$
14A5	$14^{2},7^{2}$	$9$	$14$	$38$	$( 1,13,25,37, 8,21,32, 2,14,27,38, 7,19,31)( 3,15,26,39, 9,20,33)( 4,18,29,41,11,23,34)( 5,16,28,40,10,24,35, 6,17,30,42,12,22,36)$
14A-5	$14^{2},7^{2}$	$9$	$14$	$38$	$( 1,31,19, 7,38,27,14, 2,32,21, 8,37,25,13)( 3,33,20, 9,39,26,15)( 4,34,23,11,41,29,18)( 5,36,22,12,42,30,17, 6,35,24,10,40,28,16)$
21A1	$21,7^{3}$	$4$	$21$	$38$	$( 1, 8,14,19,25,32,38)( 2, 7,13,21,27,31,37)( 3, 9,15,20,26,33,39)( 4,10,16,23,28,36,41, 5,12,18,22,30,34,42, 6,11,17,24,29,35,40)$
21A-1	$21,7^{3}$	$4$	$21$	$38$	$( 1,38,32,25,19,14, 8)( 2,37,31,27,21,13, 7)( 3,39,33,26,20,15, 9)( 4,42,36,29,22,16,11, 5,40,34,28,24,18,10, 6,41,35,30,23,17,12)$
21A2	$21,7^{3}$	$4$	$21$	$38$	$( 1,14,25,38, 8,19,32)( 2,13,27,37, 7,21,31)( 3,15,26,39, 9,20,33)( 4,16,28,41,12,22,34, 6,17,29,40,10,23,36, 5,18,30,42,11,24,35)$
21A-2	$21,7^{3}$	$4$	$21$	$38$	$( 1,32,19, 8,38,25,14)( 2,31,21, 7,37,27,13)( 3,33,20, 9,39,26,15)( 4,36,22,11,40,28,18, 6,35,23,12,42,29,16, 5,34,24,10,41,30,17)$
21A4	$21,7^{3}$	$4$	$21$	$38$	$( 1,25, 8,32,14,38,19)( 2,27, 7,31,13,37,21)( 3,26, 9,33,15,39,20)( 4,28,12,34,17,40,23, 5,30,11,35,16,41,22, 6,29,10,36,18,42,24)$
21A-4	$21,7^{3}$	$4$	$21$	$38$	$( 1,19,38,14,32, 8,25)( 2,21,37,13,31, 7,27)( 3,20,39,15,33, 9,26)( 4,22,40,18,35,12,29, 5,24,41,17,36,11,28, 6,23,42,16,34,10,30)$
21B1	$21^{2}$	$4$	$21$	$40$	$( 1,21,39,14,31, 9,25, 2,20,38,13,33, 8,27, 3,19,37,15,32, 7,26)( 4,24,42,18,36,10,29, 6,22,41,16,35,11,30, 5,23,40,17,34,12,28)$
21B-1	$21^{2}$	$4$	$21$	$40$	$( 1,27, 9,32,13,39,19, 2,26, 8,31,15,38,21, 3,25, 7,33,14,37,20)( 4,30,10,34,16,42,23, 6,28,11,36,17,41,24, 5,29,12,35,18,40,22)$
21B2	$21^{2}$	$4$	$21$	$40$	$( 1,37,33,25,21,15, 8, 2,39,32,27,20,14, 7, 3,38,31,26,19,13, 9)( 4,40,35,29,24,17,11, 6,42,34,30,22,18,12, 5,41,36,28,23,16,10)$
21B-2	$21^{2}$	$4$	$21$	$40$	$( 1, 9,13,19,26,31,38, 3, 7,14,20,27,32,39, 2, 8,15,21,25,33,37)( 4,12,17,23,30,35,41, 6,10,18,24,28,34,40, 5,11,16,22,29,36,42)$
21B4	$21^{2}$	$4$	$21$	$40$	$( 1,31,20, 8,37,26,14, 2,33,19, 7,39,25,13, 3,32,21, 9,38,27,15)( 4,36,22,11,40,28,18, 6,35,23,12,42,29,16, 5,34,24,10,41,30,17)$
21B-4	$21^{2}$	$4$	$21$	$40$	$( 1,13,26,38, 7,20,32, 2,15,25,37, 9,19,31, 3,14,27,39, 8,21,33)( 4,17,30,41,10,24,34, 5,16,29,42,12,23,35, 6,18,28,40,11,22,36)$
28A1	$28,14$	$9$	$28$	$40$	$( 1,42,37,36,32,28,27,24,19,17,13,12, 8, 5, 2,40,38,35,31,30,25,22,21,16,14,10, 7, 6)( 3,41,39,34,33,29,26,23,20,18,15,11, 9, 4)$
28A-1	$28,14$	$9$	$28$	$40$	$( 1, 6, 7,10,14,16,21,22,25,30,31,35,38,40, 2, 5, 8,12,13,17,19,24,27,28,32,36,37,42)( 3, 4, 9,11,15,18,20,23,26,29,33,34,39,41)$
28A3	$28,14$	$9$	$28$	$40$	$( 1,35,25,17, 8,42,32,22,14, 5,38,28,19,10)( 2,34,26,16, 7,41,33,24,13, 4,39,30,21,11, 3,36,27,18, 9,40,31,23,15, 6,37,29,20,12)$
28A-3	$28,14$	$9$	$28$	$40$	$( 1,10,21,30,38, 5,13,24,32,42, 7,16,25,35, 2,12,19,28,37, 6,14,22,31,40, 8,17,27,36)( 3,11,20,29,39, 4,15,23,33,41, 9,18,26,34)$
28A5	$28,14$	$9$	$28$	$40$	$( 1,30,14,40,25,12,38,24, 8,36,19, 6,32,16)( 2,29,15,42,27,11,39,22, 7,34,20, 5,31,18, 3,28,13,41,26,10,37,23, 9,35,21, 4,33,17)$
28A-5	$28,14$	$9$	$28$	$40$	$( 1,16,31, 5,19,36, 7,22,38,12,27,42,14,30, 2,17,32, 6,21,35, 8,24,37,10,25,40,13,28)( 3,18,33, 4,20,34, 9,23,39,11,26,41,15,29)$
28A9	$28,14$	$9$	$28$	$40$	$( 1,17,31, 6,19,35, 7,24,38,10,27,40,14,28, 2,16,32, 5,21,36, 8,22,37,12,25,42,13,30)( 3,18,33, 4,20,34, 9,23,39,11,26,41,15,29)$
28A-9	$28,14$	$9$	$28$	$40$	$( 1,29,15,42,25,11,39,22, 8,34,20, 5,32,18, 3,28,14,41,26,10,38,23, 9,35,19, 4,33,17)( 2,30,13,40,27,12,37,24, 7,36,21, 6,31,16)$
28A11	$28,14$	$9$	$28$	$40$	$( 1,12,21,28,38, 6,13,22,32,40, 7,17,25,36, 2,10,19,30,37, 5,14,24,31,42, 8,16,27,35)( 3,11,20,29,39, 4,15,23,33,41, 9,18,26,34)$
28A-11	$28,14$	$9$	$28$	$40$	$( 1,34,26,16, 8,41,33,24,14, 4,39,30,19,11, 3,36,25,18, 9,40,32,23,15, 6,38,29,20,12)( 2,35,27,17, 7,42,31,22,13, 5,37,28,21,10)$
28A13	$28,14$	$9$	$28$	$40$	$( 1, 5, 7,12,14,17,21,24,25,28,31,36,38,42, 2, 6, 8,10,13,16,19,22,27,30,32,35,37,40)( 3, 4, 9,11,15,18,20,23,26,29,33,34,39,41)$
28A-13	$28,14$	$9$	$28$	$40$	$( 1,40,37,35,32,30,27,22,19,16,13,10, 8, 6, 2,42,38,36,31,28,25,24,21,17,14,12, 7, 5)( 3,41,39,34,33,29,26,23,20,18,15,11, 9, 4)$

Malle's constant $a(G)$:     $1/14$

Character table

42 x 42 character table

Regular extensions

Data not computed