Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(2\) |
| Generators: |
$c, d$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_2^2:D_{99}$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $D_{99}$ |
| Order: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Automorphism Group: | $D_{99}:C_{30}$, of order \(5940\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{66}).(C_{30}\times S_3)$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_{66}$ | ||
| Normalizer: | $C_2^2:D_{99}$ | ||
| Complements: | $D_{99}$ | ||
| Minimal over-subgroups: | $C_2\times C_{22}$ | $C_2\times C_6$ | $D_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_2^2:D_{99}$ |