Subgroup ($H$) information
| Description: | $C_{22}:C_{24}$ |
| Order: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Generators: |
$abc^{195}, c^{24}, c^{176}, bc^{198}, c^{66}, c^{132}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}:C_5.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{22}.(C_2^4\times C_{10})$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^4\times F_{11}$, of order \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{22}$, of order \(44\)\(\medspace = 2^{2} \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{22}$ |