Subgroup ($H$) information
| Description: | $C_7^3:C_3^2$ |
| Order: | \(3087\)\(\medspace = 3^{2} \cdot 7^{3} \) |
| Index: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$e^{14}f^{12}g^{6}, e^{6}f^{2}g^{10}, f^{2}g^{10}, d^{2}f^{6}g^{4}, g^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $D_7^3.C_6^2:D_6$ |
| Order: | \(1185408\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2\wr S_3$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $A_4^3.C_2^3$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Outer Automorphisms: | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_2^4:\He_3.C_6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $F_7\wr S_3$, of order \(444528\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |