Subgroup ($H$) information
| Description: | $C_3\times D_{10}$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{2}, c, b^{12}, b^{30}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $D_{10}.S_3^2$ |
| Order: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| 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: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times D_6^2$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_5$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| $W$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-6$ |
| Projective image | $D_5.S_3^2$ |