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Properties

Label 4608.v.32._.J
Order $ 2^{4} \cdot 3^{2} $
Index $ 2^{5} $
Normal Yes

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Subgroup ($H$) information

Description:$C_3\times D_{24}$
Order: \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Index: \(32\)\(\medspace = 2^{5} \)
Exponent: \(24\)\(\medspace = 2^{3} \cdot 3 \)
Generators: $\left(\begin{array}{rr} 186 & 0 \\ 0 & 55 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 84 & 0 \\ 0 & 108 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 109 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 84 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 130 \end{array}\right)$ Copy content Toggle raw display
Derived length: $2$

The subgroup is characteristic (hence normal), nonabelian, and metacyclic (hence solvable, supersolvable, monomial, and metabelian).

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: $C_{32}$
Order: \(32\)\(\medspace = 2^{5} \)
Exponent: \(32\)\(\medspace = 2^{5} \)
Automorphism Group: $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \)
Outer Automorphisms: $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \)
Derived length: $1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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)$
$\operatorname{Aut}(H)$ $C_{24}:C_2^4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_{24}:C_2^4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(32\)\(\medspace = 2^{5} \)
$W$$D_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)

Related subgroups

Centralizer:$C_{192}$
Normalizer:$D_{24}.C_{96}$
Minimal over-subgroups:$D_{24}:C_6$
Maximal under-subgroups:$C_3\times D_{12}$$C_3\times D_{12}$$C_3\times C_{24}$$D_{24}$$C_3\times D_8$
Autjugate 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Other information

Möbius function$0$
Projective image$C_3 \rtimes (D_4\times C_{32})$