Subgroup ($H$) information
| Description: | $C_2\times C_7^3:S_4$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$a^{3}d^{12}e^{7}f^{2}, d^{7}e^{8}f^{6}, e^{2}f^{4}, b^{2}de^{2}f^{6}, d^{2}f, f, b^{3}e^{7}f^{3}, cd^{4}e^{7}f^{3}$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $D_7^3:(C_6\times S_3)$ |
| Order: | \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.A_4.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_7^3:(C_6\times S_3)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| $W$ | $C_7^3:(C_6\times S_4)$, of order \(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_7^3:(C_6\times S_4)$ |