Subgroup ($H$) information
| Description: | $C_5^3$ |
| Order: | \(125\)\(\medspace = 5^{3} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(5\) |
| Generators: |
$b^{9}, d, c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $5$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_5^2:C_{45}:C_4$ |
| Order: | \(4500\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_9:C_4$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.C_9.C_6.C_2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(3,5)$, of order \(1488000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \cdot 31 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times C_4^2$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1125\)\(\medspace = 3^{2} \cdot 5^{3} \) |
| $W$ | $C_3:C_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_5^2:C_{45}:C_4$ |