Subgroup ($H$) information
| Description: | $C_{39}$ |
| Order: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Generators: |
$b^{104}, b^{24}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,13$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{312}:C_4$ |
| Order: | \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| Exponent: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4\times C_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Outer Automorphisms: | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| 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), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{78}.(C_2^4\times C_{12})$, of order \(14976\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{312}$ | ||
| Normalizer: | $C_{312}:C_4$ | ||
| Complements: | $C_4\times C_8$ | ||
| Minimal over-subgroups: | $C_{78}$ | $C_3\times D_{13}$ | $C_3\times D_{13}$ |
| Maximal under-subgroups: | $C_{13}$ | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{312}:C_4$ |