Subgroup ($H$) information
| Description: | $D_{22}:C_{24}$ |
| Order: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Index: | $1$ |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Generators: |
$a, c^{132}, c^{176}, b, c^{33}, c^{24}, c^{66}$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_{22}:C_{24}$ |
| Order: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}:C_5.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{11}:C_5.C_2^6.C_2$ |
| $W$ | $D_{22}$, of order \(44\)\(\medspace = 2^{2} \cdot 11 \) |
Related subgroups
| Centralizer: | $C_2\times C_{12}$ | ||||
| Normalizer: | $D_{22}:C_{24}$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $C_{12}\times D_{22}$ | $C_2\times C_{264}$ | $C_{22}:C_{24}$ | $D_{22}:C_8$ | $C_2^2:C_{24}$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_{22}$ |