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Properties

Label 4608.v.6._.A
Order $ 2^{8} \cdot 3 $
Index $ 2 \cdot 3 $
Normal Yes

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Subgroup ($H$) information

Description:$D_{12}.C_{32}$
Order: \(768\)\(\medspace = 2^{8} \cdot 3 \)
Index: \(6\)\(\medspace = 2 \cdot 3 \)
Exponent: \(192\)\(\medspace = 2^{6} \cdot 3 \)
Generators: $\left(\begin{array}{rr} 46 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 186 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 84 & 0 \\ 0 & 108 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 109 \end{array}\right), \left(\begin{array}{rr} 0 & 84 \\ 130 & 0 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 191 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 34 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 192 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 130 \end{array}\right)$ Copy content Toggle raw display
Derived length: $2$

The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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: $C_6$
Order: \(6\)\(\medspace = 2 \cdot 3 \)
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
Automorphism Group: $C_2$, of order \(2\)
Outer Automorphisms: $C_2$, of order \(2\)
Derived length: $1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

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_3:((C_4\times C_8).C_2^6.C_2)$
$\operatorname{Aut}(H)$ $C_2\times C_{16}\times D_4\times S_3$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
$\operatorname{res}(S)$$C_3 \rtimes ((C_2\times C_8) . C_2^5)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(4\)\(\medspace = 2^{2} \)
$W$$D_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)

Related subgroups

Centralizer:$C_{192}$
Normalizer:$D_{24}.C_{96}$
Complements:$C_6$
Minimal over-subgroups:$D_{12}.C_{96}$$C_{192}.D_4$
Maximal under-subgroups:$C_{32}.D_6$$C_2\times C_{192}$$C_3:\OD_{128}$$S_3\times C_{64}$$C_{192}:C_2$$\OD_{128}:C_2$
Autjugate 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Other information

Möbius function$1$
Projective image$C_3\times D_{12}$