Subgroup ($H$) information
| Description: | $C_2\times C_{18}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,2)(7,9,10)(8,13,15)(11,12,14), (1,2), (1,2)(7,11,8,9,12,13,10,14,15), (3,6)(4,5)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 \(96768\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $56$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times {}^2G(2,3)$ |