Subgroup ($H$) information
| Description: | $C_{33}$ |
| Order: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Index: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Generators: |
$b^{154}, b^{21}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{77}:C_{204}$ |
| Order: | \(15708\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \) |
| Exponent: | \(15708\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_7:C_{68}$ |
| Order: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Automorphism Group: | $D_{14}:C_{48}$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{77}.C_{240}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{7854}$ | ||
| Normalizer: | $C_{77}:C_{204}$ | ||
| Complements: | $C_7:C_{68}$ | ||
| Minimal over-subgroups: | $C_{561}$ | $C_{231}$ | $C_{66}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{1309}:C_4$ |