Subgroup ($H$) information
| Description: | $C_{15}:D_{15}$ |
| Order: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$ab, c^{10}, d^{10}, d^{3}, c^{3}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^2:C_2^2$ |
| Order: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:S_3.C_{10}^2.C_{12}.C_4.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_4\times \AGL(2,3)\times F_5$ |
| $\card{\operatorname{res}(S)}$ | \(34560\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5\) |
| $W$ | $C_{15}:D_6$, of order \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_5$ | |||
| Normalizer: | $C_{15}^2:C_2^2$ | |||
| Complements: | $C_2$ $C_2$ | |||
| Minimal over-subgroups: | $C_{15}^2:C_2^2$ | |||
| Maximal under-subgroups: | $C_{15}^2$ | $C_5\times D_{15}$ | $C_3:D_{15}$ | $C_{15}:S_3$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_{15}^2:C_2^2$ |