Subgroup ($H$) information
| Description: | $C_{15}:D_5$ |
| Order: | \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$b^{3}, c^{3}d^{6}, d^{3}, c^{10}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^2:D_6$ |
| Order: | \(2700\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_3:S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Outer Automorphisms: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_5\times C_{10}):\GL(2,5)$, of order \(24000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_{25}:C_2^2$, of order \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_5^2:D_6$, of order \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $-27$ |
| Projective image | $(C_5\times C_{15}):D_6$ |