Subgroup ($H$) information
| Description: | $C_2\times C_{12}$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$c, d^{30}, d^{40}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_2\times C_{20}\times D_{12}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_{10}$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times \GL(3,2)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_4\times \GL(3,2)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^9.C_2^6)$ |
| $\operatorname{Aut}(H)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{W}$ | \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |