Subgroup ($H$) information
| Description: | $C_{15}^2:D_4:C_2$ |
| Order: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$a, b^{2}, d^{12}e, cd^{48}e, d^{40}, e, b^{3}d^{30}, c^{2}d^{24}e^{2}$
|
| Derived length: | $3$ |
The subgroup is normal, a semidirect factor, nonabelian, and monomial (hence solvable).
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: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_2^2.C_2^4.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{15}^2.C_4.C_2.C_2^4$ |
| $W$ | $F_5^2:S_3^2$, of order \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $F_5^2:S_3^2$ |