Properties

Label 3072.z.16.h1
Order $ 2^{6} \cdot 3 $
Index $ 2^{4} $
Normal No

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Subgroup ($H$) information

Description:$C_4\times C_{48}$
Order: \(192\)\(\medspace = 2^{6} \cdot 3 \)
Index: \(16\)\(\medspace = 2^{4} \)
Exponent: \(48\)\(\medspace = 2^{4} \cdot 3 \)
Generators: $\left(\begin{array}{rr} 29 & 30 \\ 10 & 19 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 17 & 16 \\ 16 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 8 \\ 24 & 15 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 3 & 21 \\ 7 & 28 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right)$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Ambient group ($G$) information

Description: $C_4^3:C_{48}$
Order: \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent: \(48\)\(\medspace = 2^{4} \cdot 3 \)
Derived length:$2$

The ambient group is nonabelian, monomial (hence solvable), and metabelian.

Automorphism information

While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$$C_4^2.(C_2^3\times C_{12}).C_2^6$
$\operatorname{Aut}(H)$ $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \)
$\operatorname{res}(S)$$C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(12\)\(\medspace = 2^{2} \cdot 3 \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_4\times C_{48}$
Normalizer:$C_4\times C_{48}$
Normal closure:$C_4^3:C_{48}$
Core:$C_4\times C_8$
Minimal over-subgroups:$C_4\times A_4\times C_{16}$
Maximal under-subgroups:$C_4\times C_{24}$$C_2\times C_{48}$$C_4\times C_{16}$

Other information

Number of subgroups in this autjugacy class$16$
Number of conjugacy classes in this autjugacy class$1$
Möbius function not computed
Projective image$C_4^2:C_6$