Subgroup ($H$) information
| Description: | $C_3\times D_{22}$ |
| Order: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$b, c^{176}, c^{24}, c^{132}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{24}:D_{22}$ |
| 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: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_2\times D_{22}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{12}$ | |||
| Normalizer: | $C_{24}:D_{22}$ | |||
| Minimal over-subgroups: | $C_{12}\times D_{11}$ | $S_3\times D_{22}$ | $C_{33}:D_4$ | |
| Maximal under-subgroups: | $C_{66}$ | $C_3\times D_{11}$ | $D_{22}$ | $C_2\times C_6$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{11}\times D_{12}$ |