Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$a, c^{30}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_{10}\times D_{60}$ |
| Order: | \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_5\times D_{30}$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $D_{30}:C_4^2$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_4^2$, of order \(32\)\(\medspace = 2^{5} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}:(C_2\times C_4^2\times C_2\wr C_2^2)$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-30$ |
| Projective image | $C_5\times D_{30}$ |