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Properties

Label 4608.v.144._.A
Order $ 2^{5} $
Index $ 2^{4} \cdot 3^{2} $
Normal Yes

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Subgroup ($H$) information

Description:$C_4\times C_8$
Order: \(32\)\(\medspace = 2^{5} \)
Index: \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Exponent: \(8\)\(\medspace = 2^{3} \)
Generators: $\left(\begin{array}{rr} 81 & 0 \\ 0 & 112 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 112 \end{array}\right), \left(\begin{array}{rr} 192 & 0 \\ 0 & 192 \end{array}\right), \left(\begin{array}{rr} 184 & 0 \\ 0 & 184 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 192 \end{array}\right)$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

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

Ambient group ($G$) information

Description: $D_{24}.C_{96}$
Order: \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent: \(192\)\(\medspace = 2^{6} \cdot 3 \)
Derived length:$2$

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

Quotient group ($Q$) structure

Description: $S_3\times C_{24}$
Order: \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Exponent: \(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism Group: $C_2^3\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \)
Outer Automorphisms: $C_2^4$, of order \(16\)\(\medspace = 2^{4} \)
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_3:((C_4\times C_8).C_2^6.C_2)$
$\operatorname{Aut}(H)$ $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_2^4$, of order \(16\)\(\medspace = 2^{4} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(768\)\(\medspace = 2^{8} \cdot 3 \)
$W$$C_2$, of order \(2\)

Related subgroups

Centralizer:$C_{12}\times C_{192}$
Normalizer:$D_{24}.C_{96}$
Minimal over-subgroups:$C_4\times C_{24}$$C_4\times C_{24}$$C_4\times C_{24}$$C_4\times C_{16}$$D_8:C_4$$C_8.C_8$
Maximal under-subgroups:$C_4^2$$C_2\times C_8$$C_2\times C_8$
Autjugate 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Other information

Möbius function$0$
Projective image$C_{24}\times D_{12}$