Subgroup ($H$) information
| Description: | $(C_2^2\times Q_8):S_3$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(7\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, d^{14}, b^{2}, b, c^{2}, c^{3}, d^{21}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $(C_2\times C_{12}).D_{28}$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times C_{42}).C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_3:(C_2\times C_2^7.C_2^4)$ |
| $\card{W}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |