Subgroup ($H$) information
| Description: | $C_2^3\wr C_2$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$b, c, f, g$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^4.C_2^5$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{16}.\PSL(2,7)$, of order \(176160768\)\(\medspace = 2^{23} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^{12}.\GL(3,2)$, of order \(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \) |
| $\card{W}$ | \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2^3\times C_4$ | ||
| Normalizer: | $C_2^4.C_2^5$ | ||
| Complements: | $C_4$ $C_4$ $C_4$ $C_4$ | ||
| Minimal over-subgroups: | $C_2^5:D_4$ | ||
| Maximal under-subgroups: | $C_2^3:D_4$ | $C_2^4:C_4$ | $C_2^6$ |
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | not computed |
| Projective image | not computed |