Subgroup ($H$) information
| Description: | $D_{24}:C_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$a, b^{24}, b^{12}, b^{3}c, b^{16}, b^{42}c^{2}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, nonabelian, supersolvable (hence solvable and 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^3.C_2^6.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $C_{24}:(C_2^4\times C_4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{W}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $D_{24}:Q_8$ | |||
| Minimal over-subgroups: | $D_{24}:Q_8$ | |||
| Maximal under-subgroups: | $C_2\times D_{24}$ | $C_{24}:C_4$ | $C_2\times C_{48}$ | $D_8:C_4$ |
| Autjugate subgroups: | 384.1865.2.b1.b1 |
Other information
| Möbius function | $-1$ |
| Projective image | not computed |