Subgroup ($H$) information
| Description: | $C_5^2:C_3^2$ |
| Order: | \(225\)\(\medspace = 3^{2} \cdot 5^{2} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$b^{2}c^{7}d^{2}, c^{3}d^{12}, d^{3}, c^{10}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, monomial (hence solvable), 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: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{25}:D_6$, of order \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| $\operatorname{res}(S)$ | $F_{25}:D_6$, of order \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $(C_5\times C_{15}):D_6$, of order \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $-6$ |
| Projective image | $(C_5\times C_{15}):D_6$ |