Subgroup ($H$) information
| Description: | $C_2^2\times C_{74}$ |
| Order: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Generators: |
$a, b^{4}, c^{2}, c^{37}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, abelian (hence metabelian and an A-group), and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_2\times C_{74}:C_8$ |
| Order: | \(1184\)\(\medspace = 2^{5} \cdot 37 \) |
| Exponent: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $Q_8:C_2^2.C_{37}.(C_{36}\times S_3)$ |
| $\operatorname{Aut}(H)$ | $C_{36}\times \PSL(2,7)$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_4\times C_{36}$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{37}:C_4$ |