Normal subgroup $(C_3\times C_9):(C_3\times C_6)\times A_4 \trianglelefteq S_4\times C_3^4.S_3^2$

• Label: 69984.jj.12.J
• Order: $ 2^{3} \cdot 3^{6} $
• Index: $ 2^{2} \cdot 3 $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(5832\)\(\medspace = 2^{3} \cdot 3^{6} \)
Index:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent:	not computed
Generators:	$d^{3}, g^{3}, e^{3}, f^{3}, b^{2}, d^{2}, e^{2}g^{3}, g^{7}, cf^{2}$
Derived length:	not computed

The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and metabelian (hence 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:	$D_6$
Order:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism Group:	$D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
Outer Automorphisms:	$C_2$, of order \(2\)
Derived length:	$2$

The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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:	$S_4\times C_3^4.S_3^2$
Complements:	$D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$ $D_6$
Minimal over-subgroups:	$(C_3^3.C_3^3):C_2\times A_4$	$S_4\times (C_3\times C_9):(C_3\times C_6)$	$\He_3:C_3^2:C_2^2\times A_4$	$C_6^2.C_3^4.C_2^2$
Maximal under-subgroups:	$A_4\times \He_3:C_3^2$	$A_4\times C_3^3.S_3$	$(C_3\times C_{18}):C_6^2$	$A_4\times C_3^3:C_6$	$C_3^3.(S_3\times A_4)$	$A_4\times \He_3.S_3$	$C_6^2.(S_3\times C_3^2)$

Other information

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