Subgroup ($H$) information
| Description: | $D_{24}:Q_8$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | $1$ |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$a, c^{2}, b^{12}, b^{3}, b^{42}c^{2}, b^{16}, b^{24}, c$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_{24}:Q_8$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^3.C_2^6.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $C_3:(C_2^3.C_2^6.C_2^3)$ |
| $\card{W}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||||||
| Normalizer: | $D_{24}:Q_8$ | |||||||
| Complements: | $C_1$ | |||||||
| Maximal under-subgroups: | $C_4\times D_{24}$ | $D_{24}:C_4$ | $D_{24}:C_4$ | $C_{24}:Q_8$ | $C_4:C_{48}$ | $C_{48}:C_4$ | $C_{48}:C_4$ | $D_8:Q_8$ |
Other information
| Möbius function | $1$ |
| Projective image | not computed |