Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $33$ | |
| Group : | $C_7:S_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,25,10,33,15,38,23,6,29,7,31,14,42,21,4,27,11,36,18,39,20)(2,26,9,34,16,37,24,5,30,8,32,13,41,22,3,28,12,35,17,40,19), (1,39,2,40)(3,37)(4,38)(5,42,6,41)(7,31,8,32)(9,35)(10,36)(11,34,12,33)(13,30)(14,29)(15,28,16,27)(17,26,18,25)(21,24,22,23) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 14: $D_{7}$ 24: $S_4$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 6: $S_4$
Degree 7: $D_{7}$
Degree 14: None
Degree 21: $D_{21}$
Low degree siblings
There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(19,20)(21,22)(25,26)(29,30)(31,32) (35,36)(37,38)(39,40)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $42$ | $4$ | $( 3, 5, 4, 6)( 7,42)( 8,41)( 9,40,10,39)(11,37,12,38)(13,32,14,31) (15,35,16,36)(17,34)(18,33)(19,26,20,25)(21,30,22,29)(23,27)(24,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $42$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7,41)( 8,42)( 9,40)(10,39)(11,38)(12,37)(13,32)(14,31) (15,36)(16,35)(17,33)(18,34)(19,26)(20,25)(21,29)(22,30)(23,28)(24,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)(19,22,23) (20,21,24)(25,28,29)(26,27,30)(31,34,36)(32,33,35)(37,40,42)(38,39,41)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 7,18,23,27,33,42)( 2, 8,17,24,28,34,41)( 3, 9,13,19,30,35,37) ( 4,10,14,20,29,36,38)( 5,12,16,22,26,32,40)( 6,11,15,21,25,31,39)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 7,18,23,27,33,42)( 2, 8,17,24,28,34,41)( 3,10,13,20,30,36,37, 4, 9,14,19, 29,35,38)( 5,11,16,21,26,31,40, 6,12,15,22,25,32,39)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1, 9,15,23,30,31,42, 3,11,18,19,25,33,37, 6, 7,13,21,27,35,39) ( 2,10,16,24,29,32,41, 4,12,17,20,26,34,38, 5, 8,14,22,28,36,40)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,11,13,23,25,35,42, 6, 9,18,21,30,33,39, 3, 7,15,19,27,31,37) ( 2,12,14,24,26,36,41, 5,10,17,22,29,34,40, 4, 8,16,20,28,32,38)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,13,25,42, 9,21,33, 3,15,27,37,11,23,35, 6,18,30,39, 7,19,31) ( 2,14,26,41,10,22,34, 4,16,28,38,12,24,36, 5,17,29,40, 8,20,32)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,15,29,42,11,20,33, 6,14,27,39,10,23,31, 4,18,25,38, 7,21,36) ( 2,16,30,41,12,19,34, 5,13,28,40, 9,24,32, 3,17,26,37, 8,22,35)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1,17,27,41, 7,24,33, 2,18,28,42, 8,23,34)( 3,13,30,37, 9,19,35) ( 4,14,29,38,10,20,36)( 5,15,26,39,12,21,32, 6,16,25,40,11,22,31)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,18,27,42, 7,23,33)( 2,17,28,41, 8,24,34)( 3,13,30,37, 9,19,35) ( 4,14,29,38,10,20,36)( 5,16,26,40,12,22,32)( 6,15,25,39,11,21,31)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,19,39,18,35,11,27, 3,21,42,13,31, 7,30, 6,23,37,15,33, 9,25) ( 2,20,40,17,36,12,28, 4,22,41,14,32, 8,29, 5,24,38,16,34,10,26)$ |
| $ 21, 21 $ | $8$ | $21$ | $( 1,21,37,18,31, 9,27, 6,19,42,15,35, 7,25, 3,23,39,13,33,11,30) ( 2,22,38,17,32,10,28, 5,20,41,16,36, 8,26, 4,24,40,14,34,12,29)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,23,42,18,33, 7,27)( 2,24,41,17,34, 8,28)( 3,19,37,13,35, 9,30) ( 4,20,38,14,36,10,29)( 5,22,40,16,32,12,26)( 6,21,39,15,31,11,25)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1,23,42,18,33, 7,27)( 2,24,41,17,34, 8,28)( 3,20,37,14,35,10,30, 4,19,38,13, 36, 9,29)( 5,21,40,15,32,11,26, 6,22,39,16,31,12,25)$ |
Group invariants
| Order: | $168=2^{3} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [168, 46] |
| Character table: |
2 3 3 2 2 . 2 2 . . . . 2 2 . . 2 2
3 1 . . . 1 1 . 1 1 1 1 . 1 1 1 1 .
7 1 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 4a 2b 3a 7a 14a 21a 21b 21c 21d 14b 7b 21e 21f 7c 14c
2P 1a 1a 2a 1a 3a 7b 7b 21d 21c 21e 21f 7c 7c 21a 21b 7a 7a
3P 1a 2a 4a 2b 1a 7c 14c 7c 7c 7a 7a 14a 7a 7b 7b 7b 14b
5P 1a 2a 4a 2b 3a 7b 14b 21c 21d 21f 21e 14c 7c 21b 21a 7a 14a
7P 1a 2a 4a 2b 3a 1a 2a 3a 3a 3a 3a 2a 1a 3a 3a 1a 2a
11P 1a 2a 4a 2b 3a 7c 14c 21e 21f 21b 21a 14a 7a 21c 21d 7b 14b
13P 1a 2a 4a 2b 3a 7a 14a 21b 21a 21d 21c 14b 7b 21f 21e 7c 14c
17P 1a 2a 4a 2b 3a 7c 14c 21f 21e 21a 21b 14a 7a 21d 21c 7b 14b
19P 1a 2a 4a 2b 3a 7b 14b 21d 21c 21e 21f 14c 7c 21a 21b 7a 14a
X.1 1 1 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 1 1 1
X.3 2 2 . . -1 2 2 -1 -1 -1 -1 2 2 -1 -1 2 2
X.4 2 2 . . 2 A A A A B B B B C C C C
X.5 2 2 . . 2 B B B B C C C C A A A A
X.6 2 2 . . 2 C C C C A A A A B B B B
X.7 2 2 . . -1 A A G H L K B B I J C C
X.8 2 2 . . -1 A A H G K L B B J I C C
X.9 2 2 . . -1 C C I J H G A A L K B B
X.10 2 2 . . -1 C C J I G H A A K L B B
X.11 2 2 . . -1 B B K L I J C C G H A A
X.12 2 2 . . -1 B B L K J I C C H G A A
X.13 3 -1 -1 1 . 3 -1 . . . . -1 3 . . 3 -1
X.14 3 -1 1 -1 . 3 -1 . . . . -1 3 . . 3 -1
X.15 6 -2 . . . D -A . . . . -B E . . F -C
X.16 6 -2 . . . E -B . . . . -C F . . D -A
X.17 6 -2 . . . F -C . . . . -A D . . E -B
A = E(7)^3+E(7)^4
B = E(7)+E(7)^6
C = E(7)^2+E(7)^5
D = 3*E(7)^3+3*E(7)^4
E = 3*E(7)+3*E(7)^6
F = 3*E(7)^2+3*E(7)^5
G = E(21)^5+E(21)^16
H = E(21)^2+E(21)^19
I = E(21)^8+E(21)^13
J = E(21)+E(21)^20
K = E(21)^10+E(21)^11
L = E(21)^4+E(21)^17
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