Subgroup ($H$) information
| Description: | $C_{185}$ |
| Order: | \(185\)\(\medspace = 5 \cdot 37 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(185\)\(\medspace = 5 \cdot 37 \) |
| Generators: |
$c^{111}, c^{5}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, cyclic (hence abelian, elementary ($p = 5,37$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $D_5\times D_{37}$ |
| Order: | \(740\)\(\medspace = 2^{2} \cdot 5 \cdot 37 \) |
| Exponent: | \(370\)\(\medspace = 2 \cdot 5 \cdot 37 \) |
| 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{185}.C_9.C_4^2$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_{36}$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times C_{36}$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(185\)\(\medspace = 5 \cdot 37 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{185}$ | ||
| Normalizer: | $D_5\times D_{37}$ | ||
| Complements: | $C_2^2$ | ||
| Minimal over-subgroups: | $D_5\times C_{37}$ | $C_5\times D_{37}$ | $D_{185}$ |
| Maximal under-subgroups: | $C_{37}$ | $C_5$ |
Other information
| Möbius function | $2$ |
| Projective image | $D_5\times D_{37}$ |