Subgroup ($H$) information
| Description: | $\He_3:D_{26}$ |
| Order: | \(1404\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \) |
| Index: | $1$ |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$a, c, b^{2}, d^{13}, b^{3}, d^{3}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and supersolvable (hence monomial).
Ambient group ($G$) information
| Description: | $\He_3:D_{26}$ |
| Order: | \(1404\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{13}\times C_3^2:\GL(2,3)$, of order \(67392\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $F_{13}\times C_3^2:\GL(2,3)$, of order \(67392\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 13 \) |
| $W$ | $C_{39}:D_6$, of order \(468\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 13 \) |
Related subgroups
| Centralizer: | $C_3$ | |||||||
| Normalizer: | $\He_3:D_{26}$ | |||||||
| Complements: | $C_1$ | |||||||
| Maximal under-subgroups: | $\He_3:C_{26}$ | $D_{13}\times \He_3$ | $C_3^2:D_{39}$ | $C_{39}:D_6$ | $C_{39}:D_6$ | $C_{39}:D_6$ | $C_{39}:D_6$ | $C_3^2:D_6$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{39}:D_6$ |