Galois group: 21T26

• Label: 21T26
• Degree: $21$
• Order: $882$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_7^2:(C_3\times S_3)$

Group action invariants

Degree $n$:		$21$
Transitive number $t$:		$26$
Group:		$C_7^2:(C_3\times S_3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,21,2,18,4,19)(3,15,6,20,5,16)(7,17)(8,12,14)(10,13,11), (1,11,15)(2,10,16)(3,9,17)(4,8,18)(5,14,19)(6,13,20)(7,12,21)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$18$:	$S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T26, 21T25, 42T143, 42T144, 42T152, 42T153, 42T154, 42T155

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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $	$18$	$7$	$( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$
	$ 7, 7, 7 $	$6$	$7$	$( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
	$ 7, 7, 7 $	$9$	$7$	$( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
	$ 7, 7, 7 $	$9$	$7$	$( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
	$ 7, 7, 7 $	$6$	$7$	$( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
	$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $	$49$	$3$	$( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
	$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $	$49$	$3$	$( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$98$	$3$	$( 1,21, 9)( 2,15, 8)( 3,16,14)( 4,17,13)( 5,18,12)( 6,19,11)( 7,20,10)$
	$ 21 $	$42$	$21$	$( 1,20,12, 2,15, 8, 3,17,11, 4,19,14, 5,21,10, 6,16,13, 7,18, 9)$
	$ 21 $	$42$	$21$	$( 1,20,10, 4,19,12, 7,18,14, 3,17, 9, 6,16,11, 2,15,13, 5,21, 8)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$14$	$3$	$( 1,20,13)( 2,15, 9)( 3,17,12)( 4,19, 8)( 5,21,11)( 6,16,14)( 7,18,10)$
	$ 21 $	$42$	$21$	$( 1,18,10, 4,16,11, 7,21,12, 3,19,13, 6,17,14, 2,15, 8, 5,20, 9)$
	$ 21 $	$42$	$21$	$( 1,18, 8, 3,19,11, 5,20,14, 7,21,10, 2,15,13, 4,16, 9, 6,17,12)$
	$ 3, 3, 3, 3, 3, 3, 3 $	$14$	$3$	$( 1,18,11)( 2,15, 9)( 3,19,14)( 4,16,12)( 5,20,10)( 6,17, 8)( 7,21,13)$
	$ 6, 6, 3, 3, 2, 1 $	$147$	$6$	$( 2, 5, 3)( 4, 6, 7)( 8,18,13,19, 9,21)(10,17)(11,20,12,16,14,15)$
	$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $	$21$	$2$	$( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$
	$ 14, 7 $	$63$	$14$	$( 1, 5, 2, 6, 3, 7, 4)( 8,21,12,17, 9,20,13,16,10,19,14,15,11,18)$
	$ 14, 7 $	$63$	$14$	$( 1, 6, 4, 2, 7, 5, 3)( 8,21,13,16,11,18, 9,20,14,15,12,17,10,19)$
	$ 6, 6, 3, 3, 2, 1 $	$147$	$6$	$( 2, 3, 5)( 4, 7, 6)( 8,20,10,16,11,21)( 9,18,14,15,13,17)(12,19)$

Group invariants

Order:		$882=2 \cdot 3^{2} \cdot 7^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		882.34
Character table:

		1A	2A	3A1	3A-1	3B1	3B-1	3C	6A1	6A-1	7A1	7A-1	7B1	7B-1	7C	14A1	14A-1	21A1	21A-1	21A2	21A-2
	Size	1	21	14	14	49	49	98	147	147	6	6	9	9	18	63	63	42	42	42	42
	2 P	1A	1A	3A-1	3A1	3B-1	3B1	3C	3B1	3B-1	7A1	7A-1	7B1	7B-1	7C	7B1	7B-1	21A2	21A-2	21A1	21A-1
	3 P	1A	2A	1A	1A	1A	1A	1A	2A	2A	7A-1	7A1	7B-1	7B1	7C	14A-1	14A1	7A1	7A-1	7A1	7A-1
	7 P	1A	2A	3A1	3A-1	3B1	3B-1	3C	6A1	6A-1	1A	1A	1A	1A	1A	2A	2A	3A1	3A-1	3A-1	3A1
	Type
882.34.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
882.34.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
882.34.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
882.34.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
882.34.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
882.34.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
882.34.2a	R	$2$	$0$	$-1$	$-1$	$2$	$2$	$-1$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
882.34.2b1	C	$2$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-1$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
882.34.2b2	C	$2$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-1$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
882.34.6a1	C	$6$	$0$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_7^{-3}+2-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+3+{\zeta }_7+{\zeta }_7^2$	$2{\zeta }_7^{-3}+2{\zeta }_7+2{\zeta }_7^2$	$-2{\zeta }_7^{-3}-2-2{\zeta }_7-2{\zeta }_7^2$	$-1$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$
882.34.6a2	C	$6$	$0$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	${\zeta }_7^{-3}+3+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}+2-{\zeta }_7-{\zeta }_7^2$	$-2{\zeta }_7^{-3}-2-2{\zeta }_7-2{\zeta }_7^2$	$2{\zeta }_7^{-3}+2{\zeta }_7+2{\zeta }_7^2$	$-1$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$
882.34.6b1	C	$6$	$0$	$3{\zeta }_{21}^{-7}$	$3{\zeta }_{21}^7$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{21}^{-10}+3-{\zeta }_{21}-{\zeta }_{21}^4-{\zeta }_{21}^8+{\zeta }_{21}^9$	${\zeta }_{21}^{-10}+2+{\zeta }_{21}+{\zeta }_{21}^4+{\zeta }_{21}^8-{\zeta }_{21}^9$	$2{\zeta }_{21}^{-10}-2+2{\zeta }_{21}+2{\zeta }_{21}^4+2{\zeta }_{21}^8-2{\zeta }_{21}^9$	$-2{\zeta }_{21}^{-10}-2{\zeta }_{21}-2{\zeta }_{21}^4-2{\zeta }_{21}^8+2{\zeta }_{21}^9$	$-1$	$0$	$0$	$-{\zeta }_{21}^{-10}+1-{\zeta }_{21}^2+{\zeta }_{21}^7-{\zeta }_{21}^8$	${\zeta }_{21}-{\zeta }_{21}^2+{\zeta }_{21}^4-{\zeta }_{21}^9$	$-{\zeta }_{21}+{\zeta }_{21}^2-{\zeta }_{21}^4-{\zeta }_{21}^7+{\zeta }_{21}^9$	${\zeta }_{21}^{-10}+{\zeta }_{21}^2+{\zeta }_{21}^8$
882.34.6b2	C	$6$	$0$	$3{\zeta }_{21}^7$	$3{\zeta }_{21}^{-7}$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{21}^{-10}+2+{\zeta }_{21}+{\zeta }_{21}^4+{\zeta }_{21}^8-{\zeta }_{21}^9$	$-{\zeta }_{21}^{-10}+3-{\zeta }_{21}-{\zeta }_{21}^4-{\zeta }_{21}^8+{\zeta }_{21}^9$	$-2{\zeta }_{21}^{-10}-2{\zeta }_{21}-2{\zeta }_{21}^4-2{\zeta }_{21}^8+2{\zeta }_{21}^9$	$2{\zeta }_{21}^{-10}-2+2{\zeta }_{21}+2{\zeta }_{21}^4+2{\zeta }_{21}^8-2{\zeta }_{21}^9$	$-1$	$0$	$0$	${\zeta }_{21}-{\zeta }_{21}^2+{\zeta }_{21}^4-{\zeta }_{21}^9$	$-{\zeta }_{21}^{-10}+1-{\zeta }_{21}^2+{\zeta }_{21}^7-{\zeta }_{21}^8$	${\zeta }_{21}^{-10}+{\zeta }_{21}^2+{\zeta }_{21}^8$	$-{\zeta }_{21}+{\zeta }_{21}^2-{\zeta }_{21}^4-{\zeta }_{21}^7+{\zeta }_{21}^9$
882.34.6b3	C	$6$	$0$	$3{\zeta }_{21}^{-7}$	$3{\zeta }_{21}^7$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{21}^{-10}+2+{\zeta }_{21}+{\zeta }_{21}^4+{\zeta }_{21}^8-{\zeta }_{21}^9$	$-{\zeta }_{21}^{-10}+3-{\zeta }_{21}-{\zeta }_{21}^4-{\zeta }_{21}^8+{\zeta }_{21}^9$	$-2{\zeta }_{21}^{-10}-2{\zeta }_{21}-2{\zeta }_{21}^4-2{\zeta }_{21}^8+2{\zeta }_{21}^9$	$2{\zeta }_{21}^{-10}-2+2{\zeta }_{21}+2{\zeta }_{21}^4+2{\zeta }_{21}^8-2{\zeta }_{21}^9$	$-1$	$0$	$0$	${\zeta }_{21}^{-10}+{\zeta }_{21}^2+{\zeta }_{21}^8$	$-{\zeta }_{21}+{\zeta }_{21}^2-{\zeta }_{21}^4-{\zeta }_{21}^7+{\zeta }_{21}^9$	${\zeta }_{21}-{\zeta }_{21}^2+{\zeta }_{21}^4-{\zeta }_{21}^9$	$-{\zeta }_{21}^{-10}+1-{\zeta }_{21}^2+{\zeta }_{21}^7-{\zeta }_{21}^8$
882.34.6b4	C	$6$	$0$	$3{\zeta }_{21}^7$	$3{\zeta }_{21}^{-7}$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{21}^{-10}+3-{\zeta }_{21}-{\zeta }_{21}^4-{\zeta }_{21}^8+{\zeta }_{21}^9$	${\zeta }_{21}^{-10}+2+{\zeta }_{21}+{\zeta }_{21}^4+{\zeta }_{21}^8-{\zeta }_{21}^9$	$2{\zeta }_{21}^{-10}-2+2{\zeta }_{21}+2{\zeta }_{21}^4+2{\zeta }_{21}^8-2{\zeta }_{21}^9$	$-2{\zeta }_{21}^{-10}-2{\zeta }_{21}-2{\zeta }_{21}^4-2{\zeta }_{21}^8+2{\zeta }_{21}^9$	$-1$	$0$	$0$	$-{\zeta }_{21}+{\zeta }_{21}^2-{\zeta }_{21}^4-{\zeta }_{21}^7+{\zeta }_{21}^9$	${\zeta }_{21}^{-10}+{\zeta }_{21}^2+{\zeta }_{21}^8$	$-{\zeta }_{21}^{-10}+1-{\zeta }_{21}^2+{\zeta }_{21}^7-{\zeta }_{21}^8$	${\zeta }_{21}-{\zeta }_{21}^2+{\zeta }_{21}^4-{\zeta }_{21}^9$
882.34.9a1	C	$9$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-3{\zeta }_7^{-3}-3-3{\zeta }_7-3{\zeta }_7^2$	$3{\zeta }_7^{-3}+3{\zeta }_7+3{\zeta }_7^2$	$-{\zeta }_7^{-3}-2-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}-1+{\zeta }_7+{\zeta }_7^2$	$2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	$0$	$0$	$0$	$0$
882.34.9a2	C	$9$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3{\zeta }_7^{-3}+3{\zeta }_7+3{\zeta }_7^2$	$-3{\zeta }_7^{-3}-3-3{\zeta }_7-3{\zeta }_7^2$	${\zeta }_7^{-3}-1+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-2-{\zeta }_7-{\zeta }_7^2$	$2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$0$	$0$	$0$	$0$
882.34.9b1	C	$9$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-3{\zeta }_7^{-3}-3-3{\zeta }_7-3{\zeta }_7^2$	$3{\zeta }_7^{-3}+3{\zeta }_7+3{\zeta }_7^2$	$-{\zeta }_7^{-3}-2-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}-1+{\zeta }_7+{\zeta }_7^2$	$2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$	$0$	$0$	$0$	$0$
882.34.9b2	C	$9$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3{\zeta }_7^{-3}+3{\zeta }_7+3{\zeta }_7^2$	$-3{\zeta }_7^{-3}-3-3{\zeta }_7-3{\zeta }_7^2$	${\zeta }_7^{-3}-1+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-2-{\zeta }_7-{\zeta }_7^2$	$2$	${\zeta }_7^{-3}+1+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-{\zeta }_7-{\zeta }_7^2$	$0$	$0$	$0$	$0$
882.34.18a	R	$18$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-3$	$-3$	$4$	$4$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$