Subgroup ($H$) information
| Description: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, b$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^2:C_{60}$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{15}$ |
| Order: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Automorphism Group: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3.C_2^5$, of order \(256\)\(\medspace = 2^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2\times C_{30}$ | ||
| Normalizer: | $C_2^2:C_{60}$ | ||
| Complements: | $C_{15}$ | ||
| Minimal over-subgroups: | $C_2^2:C_{20}$ | $C_2^2:C_{12}$ | |
| Maximal under-subgroups: | $C_2^3$ | $C_2\times C_4$ | $C_2\times C_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_2\times C_{30}$ |