Subgroup ($H$) information
| Description: | $C_{13}^2:(C_4\times S_3)$ |
| Order: | \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
| Index: | $1$ |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Generators: |
$a, d, b^{6}c^{4}d^{11}, b^{4}c^{7}d^{3}, b^{3}, cd^{7}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, monomial, and an A-group.
Ambient group ($G$) information
| Description: | $C_{13}^2:(C_4\times S_3)$ |
| Order: | \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{13}^2.(C_6\times S_3)$, of order \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $D_{13}^2.(C_6\times S_3)$, of order \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $W$ | $C_{13}^2:(C_4\times S_3)$, of order \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $C_{13}^2:(C_4\times S_3)$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $C_{13}^2:D_6$ | $C_{13}^2:C_{12}$ | $C_{13}^2:(C_3:C_4)$ | $D_{13}^2.C_2$ | $C_4\times S_3$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{13}^2:(C_4\times S_3)$ |