Subgroup ($H$) information
| Description: | $C_{15}^2:C_2^2$ |
| Order: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$ab^{3}d^{24}e^{2}, e, b^{2}, d^{40}, c^{2}d^{12}e^{3}, d^{12}e$
|
| 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: | $F_5^2:S_3^2$ |
| Order: | \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_2^2.C_2^4.C_2^2$ |
| $\operatorname{Aut}(H)$ | $\AGL(2,3)\times C_5^2:C_4.S_5$ |
| $W$ | $F_5^2:S_3^2$, of order \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $F_5^2:S_3^2$ |