Subgroup ($H$) information
| Description: | $C_4^4.C_{24}$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Index: | $1$ |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
29 & 30 \\
10 & 19
\end{array}\right), \left(\begin{array}{rr}
7 & 24 \\
8 & 15
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
16 & 1
\end{array}\right), \left(\begin{array}{rr}
3 & 21 \\
7 & 28
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
16 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 8 \\
8 & 1
\end{array}\right), \left(\begin{array}{rr}
31 & 16 \\
16 & 31
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 21
\end{array}\right), \left(\begin{array}{rr}
9 & 8 \\
16 & 25
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and metabelian. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_4^4.C_{24}$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2.(C_2^4\times C_{12}).C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4^2.(C_2^4\times C_{12}).C_2^6.C_2^3$ |
| $W$ | $C_4^2:C_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_4\times C_8$ | ||||
| Normalizer: | $C_4^4.C_{24}$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $C_4^4.C_{12}$ | $C_2^2\times C_4^2:C_{48}$ | $C_4^3:C_{48}$ | $C_4^4.C_8$ | $C_2^3:C_{12}\times C_{16}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_4^2:C_6$ |