Subgroup ($H$) information
| Description: | $D_4^2:S_3$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(3\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, cd^{6}, d^{6}, b^{3}, b^{6}, d^{4}, c^{2}, d^{3}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, supersolvable (hence solvable and monomial), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{12}^2:D_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| 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), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^4.C_2^5.C_2^2)$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_2^3.D_4^2$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times C_2^3.D_4^2$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $(C_2\times C_{12}):D_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $(C_6\times C_{12}):D_4$ |