Normal subgroup $C_6^2.C_3^3.C_3.D_6 \trianglelefteq S_4\times C_3^4.S_3^2$

• Label: 69984.jj.2.G
• Order: $ 2^{4} \cdot 3^{7} $
• Index: $ 2 $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Index:	\(2\)
Exponent:	not computed
Generators:	$ad^{3}, g^{3}, f^{2}, g^{7}, e^{3}f^{3}, b^{3}d^{3}, c, e^{2}, d^{2}, f^{3}, b^{2}$
Derived length:	not computed

The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.

Ambient group ($G$) information

Description:	$S_4\times C_3^4.S_3^2$
Order:	\(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Derived length:	$3$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

Quotient group ($Q$) structure

Description:	$C_2$
Order:	\(2\)
Exponent:	\(2\)
Automorphism Group:	$C_1$, of order $1$
Outer Automorphisms:	$C_1$, of order $1$
Derived length:	$1$

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

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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$S_4\times C_3^4.S_3^2$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
$\operatorname{Aut}(H)$	not computed
$W$	$C_{2022}:C_{28}$, of order \(56616\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 337 \)

Related subgroups

Centralizer:	$C_1$
Normalizer:	$S_4\times C_3^4.S_3^2$
Complements:	$C_2$ $C_2$ $C_2$ $C_2$ $C_2$ $C_2$
Minimal over-subgroups:	$S_4\times C_3^4.S_3^2$
Maximal under-subgroups:	$(C_3^3.C_3^3):C_2\times A_4$	$(C_3\times A_4).C_3^4:C_6$	$C_6^2.C_3^4.C_6$	$C_6^2.C_3^3.D_6$	$C_6^2.C_3^4.C_2^2$	$(C_9\times A_4).C_3^3.C_2^2$	$C_6^2.C_3^4.C_2^2$	$C_6^2.C_3^4.C_2^2$	$C_3^4.C_3.C_6.C_2^3$	$C_3^4.C_3^2.D_6$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	$S_4\times C_3^4.S_3^2$