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: |
$ab^{2}c^{27}, c^{24}, c^{36}, b, c^{16}, b^{2}c^{30}, c^{30}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_2^3.D_{24}$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:((C_2\times C_4^2).C_2^6)$ |
| $\operatorname{Aut}(H)$ | $C_{24}:(C_2^4\times C_4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{24}:(C_2^4\times C_4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times D_{24}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_2^3.D_{24}$ | |||
| Complements: | $C_2$ | |||
| Minimal over-subgroups: | $C_2^3.D_{24}$ | |||
| Maximal under-subgroups: | $C_2\times D_{24}$ | $C_{24}:C_4$ | $C_2\times C_{48}$ | $D_8:C_4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{24}$ |