Subgroup ($H$) information
| Description: | $C_2^5$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(5\) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,2)(9,10), (1,2)(7,8), (5,6), (1,2)(3,4), (1,2)(5,6)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, maximal, a semidirect factor, abelian (hence metabelian and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2\wr C_5$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| 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), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{16}:C_4$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $\GL(5,2)$, of order \(9999360\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{15}:C_4$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_5$, of order \(5\) |
Related subgroups
| Centralizer: | $C_2^5$ | ||||||
| Normalizer: | $C_2\wr C_5$ | ||||||
| Complements: | $C_5$ | ||||||
| Minimal over-subgroups: | $C_2\wr C_5$ | ||||||
| Maximal under-subgroups: | $C_2^4$ | $C_2^4$ | $C_2^4$ | $C_2^4$ | $C_2^4$ | $C_2^4$ | $C_2^4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2^4:C_5$ |