Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(3\) |
| Generators: |
$c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $(C_2\times C_{24}):D_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $\OD_{16}:D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $(C_2^6\times C_4).C_2^5$, of order \(8192\)\(\medspace = 2^{13} \) |
| Outer Automorphisms: | $C_2^5:C_2^3$, of order \(256\)\(\medspace = 2^{8} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2\times C_2^6.C_2^6.C_2)$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | \(2\) |
Related subgroups
| Centralizer: | $C_{24}:C_2^3$ | ||||
| Normalizer: | $(C_2\times C_{24}):D_4$ | ||||
| Complements: | $\OD_{16}:D_4$ | ||||
| Minimal over-subgroups: | $C_6$ | $C_6$ | $C_6$ | $C_6$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |