Group action invariants
| Degree $n$ : | $28$ | |
| Transitive number $t$ : | $30$ | |
| Group : | $C_7:S_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,5,21,9,25,13)(2,20,7,22,12,27,14,4,19,6,24,11,26,16,3,18,8,23,10,28,15), (1,10,4,11)(2,12,3,9)(5,6,8,7)(13,26,16,27)(14,28,15,25)(17,22,20,23)(18,24,19,21) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 14: $D_{7}$ 24: $S_4$ 42: $D_{21}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 7: $D_{7}$
Degree 14: None
Low degree siblings
42T32, 42T33Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $42$ | $2$ | $( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)(13,17)(14,18) (15,20)(16,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)(22,23,24)(26,27,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4 $ | $42$ | $4$ | $( 1, 2, 3, 4)( 5,26, 7,28)( 6,27, 8,25)( 9,22,11,24)(10,23,12,21)(13,18,15,20) (14,19,16,17)$ |
| $ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27) ( 4, 8,12,16,20,24,28)$ |
| $ 21, 7 $ | $8$ | $21$ | $( 1, 5, 9,13,17,21,25)( 2, 7,12,14,19,24,26, 3, 8,10,15,20,22,27, 4, 6,11,16, 18,23,28)$ |
| $ 21, 7 $ | $8$ | $21$ | $( 1, 5, 9,13,17,21,25)( 2, 8,11,14,20,23,26, 4, 7,10,16,19,22,28, 3, 6,12,15, 18,24,27)$ |
| $ 14, 14 $ | $6$ | $14$ | $( 1, 6, 9,14,17,22,25, 2, 5,10,13,18,21,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$ |
| $ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23) ( 4,12,20,28, 8,16,24)$ |
| $ 21, 7 $ | $8$ | $21$ | $( 1, 9,17,25, 5,13,21)( 2,11,20,26, 7,16,22, 3,12,18,27, 8,14,23, 4,10,19,28, 6,15,24)$ |
| $ 21, 7 $ | $8$ | $21$ | $( 1, 9,17,25, 5,13,21)( 2,12,19,26, 8,15,22, 4,11,18,28, 7,14,24, 3,10,20,27, 6,16,23)$ |
| $ 14, 14 $ | $6$ | $14$ | $( 1,10,17,26, 5,14,21, 2, 9,18,25, 6,13,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$ |
| $ 7, 7, 7, 7 $ | $2$ | $7$ | $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19) ( 4,16,28,12,24, 8,20)$ |
| $ 21, 7 $ | $8$ | $21$ | $( 1,13,25, 9,21, 5,17)( 2,15,28,10,23, 8,18, 3,16,26,11,24, 6,19, 4,14,27,12, 22, 7,20)$ |
| $ 21, 7 $ | $8$ | $21$ | $( 1,13,25, 9,21, 5,17)( 2,16,27,10,24, 7,18, 4,15,26,12,23, 6,20, 3,14,28,11, 22, 8,19)$ |
| $ 14, 14 $ | $6$ | $14$ | $( 1,14,25,10,21, 6,17, 2,13,26, 9,22, 5,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$ |
Group invariants
| Order: | $168=2^{3} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [168, 46] |
| Character table: |
2 3 2 . 3 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 3a 2b 4a 7a 21a 21b 14a 7b 21c 21d 14b 7c 21e 21f 14c
2P 1a 1a 3a 1a 2b 7b 21d 21c 7b 7c 21e 21f 7c 7a 21a 21b 7a
3P 1a 2a 1a 2b 4a 7c 7c 7c 14c 7a 7a 7a 14a 7b 7b 7b 14b
5P 1a 2a 3a 2b 4a 7b 21c 21d 14b 7c 21f 21e 14c 7a 21b 21a 14a
7P 1a 2a 3a 2b 4a 1a 3a 3a 2b 1a 3a 3a 2b 1a 3a 3a 2b
11P 1a 2a 3a 2b 4a 7c 21e 21f 14c 7a 21b 21a 14a 7b 21c 21d 14b
13P 1a 2a 3a 2b 4a 7a 21b 21a 14a 7b 21d 21c 14b 7c 21f 21e 14c
17P 1a 2a 3a 2b 4a 7c 21f 21e 14c 7a 21a 21b 14a 7b 21d 21c 14b
19P 1a 2a 3a 2b 4a 7b 21d 21c 14b 7c 21e 21f 14c 7a 21a 21b 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 . -1 2 . 2 -1 -1 2 2 -1 -1 2 2 -1 -1 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 . -1 2 . A G H A B L K B C I J C
X.8 2 . -1 2 . A H G A B K L B C J I C
X.9 2 . -1 2 . C I J C A H G A B L K B
X.10 2 . -1 2 . C J I C A G H A B K L B
X.11 2 . -1 2 . B K L B C I J C A G H A
X.12 2 . -1 2 . B L K B C J I C A H G A
X.13 3 -1 . -1 1 3 . . -1 3 . . -1 3 . . -1
X.14 3 1 . -1 -1 3 . . -1 3 . . -1 3 . . -1
X.15 6 . . -2 . D . . -A E . . -B F . . -C
X.16 6 . . -2 . E . . -B F . . -C D . . -A
X.17 6 . . -2 . F . . -C D . . -A 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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