Subgroup ($H$) information
| Description: | $C_{55}:C_5$ |
| Order: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$b^{2}, c^{20}, c^{44}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 5$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{44}.C_{10}^2$ |
| Order: | \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times Q_8$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_{165}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $F_5\times F_{11}$, of order \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_5\times F_{11}$, of order \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_5\times Q_8$ | ||
| Normalizer: | $C_{44}.C_{10}^2$ | ||
| Complements: | $C_2\times Q_8$ | ||
| Minimal over-subgroups: | $C_{110}:C_5$ | $C_5\times F_{11}$ | |
| Maximal under-subgroups: | $C_{55}$ | $C_{11}:C_5$ | $C_5^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $Q_8\times F_{11}$ |