Subgroup ($H$) information
| Description: | $C_7^3:(C_6^2:A_4)$ |
| Order: | \(148176\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$f^{2}g^{10}, e^{6}, e^{14}f^{12}g^{6}, e^{21}f^{5}g^{5}, c, g^{7}, b^{2}cde^{21}f^{3}g^{6}, d^{2}e^{24}g^{2}, d^{3}e^{24}g^{10}, g^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial 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: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_2^4:\He_3.C_6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_7^3.(C_2^3\times C_6).A_4.\He_3.C_2^2$ |
| $\card{W}$ | 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 |