Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \) |
| Exponent: | \(2\) |
| Generators: |
$a, b, d^{38}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_{38}:C_6\times Q_8$ |
| Order: | \(1824\)\(\medspace = 2^{5} \cdot 3 \cdot 19 \) |
| Exponent: | \(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{38}:C_6$ |
| Order: | \(228\)\(\medspace = 2^{2} \cdot 3 \cdot 19 \) |
| Exponent: | \(114\)\(\medspace = 2 \cdot 3 \cdot 19 \) |
| Automorphism Group: | $S_3\times F_{19}$, of order \(2052\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 19 \) |
| Outer Automorphisms: | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^4.C_{57}.C_3.C_9.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(131328\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 19 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{38}:C_6\times Q_8$ | ||
| Normalizer: | $C_{38}:C_6\times Q_8$ | ||
| Minimal over-subgroups: | $C_2^2\times C_{38}$ | $C_2^2\times C_6$ | $C_2^2\times C_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $38$ |
| Projective image | $C_{38}:C_6$ |