Subgroup ($H$) information
| Description: | $D_{23}^2:D_{11}$ |
| Order: | \(46552\)\(\medspace = 2^{3} \cdot 11 \cdot 23^{2} \) |
| Index: | \(11\) |
| Exponent: | \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \) |
| Generators: |
$b^{2}c^{308}d^{3}, c^{22}d^{8}, d, b^{11}c^{198}d^{9}, c^{253}, ab^{8}c^{236}d^{5}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $F_{23}\wr C_2$ |
| Order: | \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| Exponent: | \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Exponent: | \(11\) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{23}\wr C_2$, of order \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| $\operatorname{Aut}(H)$ | $F_{23}\wr C_2$, of order \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| $W$ | $F_{23}\wr C_2$, of order \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $F_{23}\wr C_2$ |