Subgroup ($H$) information
| Description: | $C_2\times \He_3$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$d^{15}, b, cd^{20}, c$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^2:D_{30}$ |
| Order: | \(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{30}:C_{12}:\GL(2,3)$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_3:S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
Other information
| Möbius function | $5$ |
| Projective image | $C_3:D_{15}$ |