Subgroup ($H$) information
| Description: | $C_2^5:D_4$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(7\) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{7}, g, bce, e, cd$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, nonabelian, a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^5:F_8$ |
| Order: | \(1792\)\(\medspace = 2^{8} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_7$ |
| Order: | \(7\) |
| Exponent: | \(7\) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6:C_7.A_4.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^{15}.C_2^4.\PSL(2,7)$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2\times F_8$, of order \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2^4$ | ||||
| Normalizer: | $C_2^5:F_8$ | ||||
| Complements: | $C_7$ | ||||
| Minimal over-subgroups: | $C_2^5:F_8$ | ||||
| Maximal under-subgroups: | $C_2^7$ | $C_2^3\wr C_2$ | $C_2^4:D_4$ | $C_2^5:C_4$ | $C_2^3\wr C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2^4:F_8$ |