Subgroup ($H$) information
| Description: | $C_7^3$ |
| Order: | \(343\)\(\medspace = 7^{3} \) |
| Index: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Exponent: | \(7\) |
| Generators: |
$e^{6}f^{2}g^{10}, g^{2}, f^{2}g^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $7$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct 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: | $C_2^2\times C_6^2:S_4$ |
| Order: | \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $(C_2^5\times C_6).C_3^4.C_2^4$ |
| Outer Automorphisms: | $S_4\times S_3^2$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_2^4:\He_3.C_6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(3,7)$, of order \(33784128\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7^{3} \cdot 19 \) |
| $\card{W}$ | 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 |