Subgroup ($H$) information
| Description: | $C_{19}$ |
| Order: | \(19\) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(19\) |
| Generators: |
$c^{4}$
|
| 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, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $19$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_4\times C_{76}:C_6$ |
| 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), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4\times C_{12}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^7:S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Outer Automorphisms: | $C_2^7:S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_{114}.C_9.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(7296\)\(\medspace = 2^{7} \cdot 3 \cdot 19 \) |
| $W$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_4\times C_{76}$ | ||
| Normalizer: | $C_4\times C_{76}:C_6$ | ||
| Complements: | $C_2\times C_4\times C_{12}$ | ||
| Minimal over-subgroups: | $C_{19}:C_3$ | $C_{38}$ | $D_{19}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_4\times C_{76}:C_6$ |