Subgroup ($H$) information
| Description: | $C_2^{12}.C_{14}$ |
| Order: | \(57344\)\(\medspace = 2^{13} \cdot 7 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(5,8)(6,7)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27), (1,2)(3,4)(5,6)(7,8) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metabelian (hence solvable), and an A-group. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.(C_7\times S_4)$ |
| Order: | \(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 \(28901376\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $F_5^3$, of order \(305892163584\)\(\medspace = 2^{19} \cdot 3^{5} \cdot 7^{4} \) |
| $W$ | $C_2^{12}.C_{14}$, of order \(57344\)\(\medspace = 2^{13} \cdot 7 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $\ASigmaL(1,16384)$ |
| Normal closure: | $C_2^7.C_2\wr C_7$ |
| Core: | $C_2^{12}:C_7$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^{12}.(C_7\times S_4)$ |