Properties

Label 384.5460.96.a1.a1
Order $ 2^{2} $
Index $ 2^{5} \cdot 3 $
Normal Yes

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Subgroup ($H$) information

Description:$C_2^2$
Order: \(4\)\(\medspace = 2^{2} \)
Index: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent: \(2\)
Generators: $b^{2}, c^{24}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

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

Ambient group ($G$) information

Description: $C_{48}:D_4$
Order: \(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent: \(48\)\(\medspace = 2^{4} \cdot 3 \)
Nilpotency class:$2$
Derived length:$2$

The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Quotient group ($Q$) structure

Description: $C_2^2\times C_{24}$
Order: \(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent: \(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism Group: $C_2^5:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Outer Automorphisms: $C_2^5:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Nilpotency class: $1$
Derived length: $1$

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

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_2^4.C_2^6$
$\operatorname{Aut}(H)$ $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_1$, of order $1$
$\card{\operatorname{ker}(\operatorname{res})}$\(1024\)\(\medspace = 2^{10} \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_{48}:D_4$
Normalizer:$C_{48}:D_4$
Minimal over-subgroups:$C_2\times C_6$$C_2^3$$C_2^3$$C_2\times C_4$$C_2\times C_4$$C_2\times C_4$$C_2\times C_4$$C_2\times C_4$
Maximal under-subgroups:$C_2$$C_2$$C_2$

Other information

Möbius function$0$
Projective image$C_2^2\times C_{24}$