Subgroup ($H$) information
| Description: | $\SOPlus(4,2)$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$b^{3}, e, c^{18}e^{2}, c^{12}de^{2}, de$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_3^4:(C_4\times \SD_{16})$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2\times C_3^2:C_4$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $F_9:C_2^2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_8^2.C_2^2$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^2:C_4$ | ||||
| Normalizer: | $C_3^4:(C_4\times \SD_{16})$ | ||||
| Minimal over-subgroups: | $S_3^2:C_6$ | $S_3^2:C_6$ | $F_9:C_2$ | $S_3^2:C_2^2$ | $F_9:C_2$ |
| Maximal under-subgroups: | $C_3^2:C_4$ | $S_3^2$ | $D_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_3^4:(C_4\times \SD_{16})$ |