Normal subgroup $S_4\times C_9:(C_3\times S_3) \trianglelefteq S_4\times C_3^4.S_3^2$

• Label: 69984.jj.18.A
• Order: $ 2^{4} \cdot 3^{5} $
• Index: $ 2 \cdot 3^{2} $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(3888\)\(\medspace = 2^{4} \cdot 3^{5} \)
Index:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Exponent:	not computed
Generators:	$a, f^{3}, g^{7}, e^{2}, d^{3}, b^{2}, e^{3}f^{3}, g^{3}, d^{2}$
Derived length:	not computed

The subgroup is characteristic (hence normal), 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:	$C_3:S_3$
Order:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism Group:	$C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Outer Automorphisms:	$S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:	$2$

The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, 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:	$S_4\times C_3^4.S_3^2$
Minimal over-subgroups:	$S_4\times C_3:D_9:C_3^2$	$S_4\times (C_9.C_3^3):C_2$	$S_4\times (C_3\times C_9):(C_3\times C_6)$	$S_4\times (\He_3.S_3):C_3$	$S_4\times S_3\times D_9:C_3$
Maximal under-subgroups:	$C_9:C_3^2\times S_4$	$A_4\times C_3^3.S_3$	$(A_4\times C_3^3).S_3$	$C_6^2.S_3^2$

Other information

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