Properties

Label 768.1088504.12.c1.a1
Order $ 2^{6} $
Index $ 2^{2} \cdot 3 $
Normal Yes

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Subgroup ($H$) information

Description:$C_2^3\times C_8$
Order: \(64\)\(\medspace = 2^{6} \)
Index: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent: \(8\)\(\medspace = 2^{3} \)
Generators: $\left(\begin{array}{rr} 21 & 16 \\ 16 & 5 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 17 & 16 \\ 0 & 17 \end{array}\right)$ 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), and a $p$-group (hence elementary and hyperelementary).

Ambient group ($G$) information

Description: $C_2\times A_4:\OD_{32}$
Order: \(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent: \(48\)\(\medspace = 2^{4} \cdot 3 \)
Derived length:$3$

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

Quotient group ($Q$) structure

Description: $C_3:C_4$
Order: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent: \(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism Group: $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
Outer Automorphisms: $C_2$, of order \(2\)
Nilpotency class: $-1$
Derived length: $2$

The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

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_2^3\times A_4).C_2^6$
$\operatorname{Aut}(H)$ $C_2.C_2^7:\GL(3,2)$, of order \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \)
$\operatorname{res}(\operatorname{Aut}(G))$$D_4\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(64\)\(\medspace = 2^{6} \)
$W$$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)

Related subgroups

Centralizer:$C_2^4\times C_8$
Normalizer:$C_2\times A_4:\OD_{32}$
Minimal over-subgroups:$C_2^3:C_{24}$$C_2^4\times C_8$
Maximal under-subgroups:$C_2^3\times C_4$$C_2^2\times C_8$$C_2^2\times C_8$$C_2^2\times C_8$$C_2^2\times C_8$$C_2^2\times C_8$$C_2^2\times C_8$

Other information

Möbius function$0$
Projective image$C_2^2.S_4$