Subgroup ($H$) information
| Description: | $C_{10}^2.C_2^2$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ad^{6}ef^{3}g, e^{2}g^{4}, cd^{6}e^{4}f^{3}, b^{2}cd^{8}f^{2}g^{2}, bdef^{3}, d^{2}e^{8}f^{4}g$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(\SO(3,7)\times S_4^2).C_2^2$ |
| Order: | \(40000\)\(\medspace = 2^{6} \cdot 5^{4} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(1280000\)\(\medspace = 2^{11} \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | $D_{10}^2.D_4^2$, of order \(25600\)\(\medspace = 2^{10} \cdot 5^{2} \) |
| $W$ | $D_5\wr C_2$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $50$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_5:D_5^3:C_2^2$ |