Subgroup ($H$) information
| Description: | $C_3^7:C_2^3$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Index: | \(5878656\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 7 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(16,18,17)(37,38,39), (13,15,14)(34,36,35), (4,6,5)(13,15,14)(16,17,18) \!\cdots\! \rangle$
|
| 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: | $C_3^{14}.C_2^4.C_2.\PSL(2,7)$ |
| Order: | \(102852965376\)\(\medspace = 2^{10} \cdot 3^{15} \cdot 7 \) |
| Exponent: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_3^7.C_2^4:\GL(3,2)$ |
| Order: | \(5878656\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 7 \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Automorphism Group: | $C_3^7.C_2^4:\GL(3,2)$, of order \(5878656\)\(\medspace = 2^{7} \cdot 3^{8} \cdot 7 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is nonabelian and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(205705930752\)\(\medspace = 2^{11} \cdot 3^{15} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_3^7.(C_2^7.\GL(3,2))$, of order \(47029248\)\(\medspace = 2^{10} \cdot 3^{8} \cdot 7 \) |
| $\card{W}$ | 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 |