Subgroup ($H$) information
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$bc, d^{129}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $C_{172}.C_2^3$ |
| Order: | \(1376\)\(\medspace = 2^{5} \cdot 43 \) |
| Exponent: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{86}$ |
| Order: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Exponent: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Automorphism Group: | $S_3\times C_{42}$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $S_3\times C_{42}$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}\times A_4^2.D_4$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $D_4\times C_{43}$ | ||
| Normalizer: | $C_{172}.C_2^3$ | ||
| Complements: | $C_2\times C_{86}$ | ||
| Minimal over-subgroups: | $D_4\times C_{43}$ | $D_4:C_2$ | $C_2\times D_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $18$ |
| Möbius function | $-2$ |
| Projective image | $C_2^3\times C_{86}$ |