Subgroup ($H$) information
| Description: | $C_2^2\times D_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(3,5)(4,6), (3,6)(4,5), (1,2)(7,12)(8,10)(9,15)(13,14), (1,2), (4,5)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $D_4\times \SL(2,8):C_6$ |
| Order: | \(24192\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,8).C_3\times C_2\wr C_2^2$, of order \(96768\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^6:(C_2\times S_4)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $63$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-8$ |
| Projective image | $C_2^2\times {}^2G(2,3)$ |