Subgroup ($H$) information
| Description: | $C_3\times D_{13}$ |
| Order: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$a^{2}b^{12}, b^{24}, b^{104}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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: | $C_2\times C_8$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{78}.(C_2^4\times C_{12})$, of order \(14976\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_{13}$, of order \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{13}$, of order \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_{13}:C_4$, of order \(52\)\(\medspace = 2^{2} \cdot 13 \) |
Related subgroups
| Centralizer: | $C_{24}$ | ||
| Normalizer: | $C_{312}:C_4$ | ||
| Minimal over-subgroups: | $C_3\times D_{26}$ | $C_{39}:C_4$ | $C_{39}:C_4$ |
| Maximal under-subgroups: | $C_{39}$ | $D_{13}$ | $C_6$ |
| Autjugate subgroups: | 1248.473.16.b1.a1 |
Other information
| Möbius function | $0$ |
| Projective image | $C_{312}:C_4$ |