Subgroup ($H$) information
| Description: | $D_{10}\times \He_3$ |
| Order: | \(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{3}, c^{3}, a^{2}, c^{10}, b^{6}, b^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{30}.C_6^2$ |
| Order: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \AGL(2,3)\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_{30}:C_{12}:\GL(2,3)$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{30}:C_{12}:\GL(2,3)$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3^2\times D_5$, of order \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_{12}$ | |||
| Normalizer: | $C_{30}.C_6^2$ | |||
| Minimal over-subgroups: | $C_{30}.C_6^2$ | |||
| Maximal under-subgroups: | $C_{10}\times \He_3$ | $D_5\times \He_3$ | $C_3^2\times D_{10}$ | $C_2^2\times \He_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^2\times D_{10}$ |