Subgroup ($H$) information
| Description: | $C_2^{12}:C_7$ |
| Order: | \(28672\)\(\medspace = 2^{12} \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(5,6)(7,8)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,10,18,26,5,15,22) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, metabelian (hence solvable), and an A-group. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.C_{14}$ |
| Order: | \(57344\)\(\medspace = 2^{13} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metabelian (hence solvable), and an A-group. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5^3$, of order \(305892163584\)\(\medspace = 2^{19} \cdot 3^{5} \cdot 7^{4} \) |
| $\operatorname{Aut}(H)$ | Group of order \(305892163584\)\(\medspace = 2^{19} \cdot 3^{5} \cdot 7^{4} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^{12}.C_{14}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^{12}.C_{14}$ |