Subgroup ($H$) information
| Description: | $C_5^2\times C_{15}$ |
| Order: | \(375\)\(\medspace = 3 \cdot 5^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$d^{10}, c, d^{3}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, abelian (hence metabelian and an A-group), and elementary for $p = 5$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_5^2:(C_3\times D_{15})$ |
| Order: | \(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.(C_6\times C_{12}).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4.\PSL(3,5)$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times C_8:D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(375\)\(\medspace = 3 \cdot 5^{3} \) |
| $W$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_5^2:(C_3\times D_{15})$ |