Normal subgroup $C_2\times C_6^4 \trianglelefteq C_2^5:(\He_3^2:C_4)$

• Label: 93312.dy.36.A
• Order: $ 2^{5} \cdot 3^{4} $
• Index: $ 2^{2} \cdot 3^{2} $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Index:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Exponent:	not computed
Generators:	$b^{3}, f^{2}, f^{3}, e^{2}g, c^{3}d^{3}, e^{3}, c^{2}d^{4}e^{5}, c^{4}d^{2}e^{4}fg, d^{3}$
Nilpotency class:	not computed
Derived length:	not computed

The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, and abelian (hence metabelian and an A-group). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.

Ambient group ($G$) information

Description:	$C_2^5:(\He_3^2:C_4)$
Order:	\(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Derived length:	$3$

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

Quotient group ($Q$) structure

Description:	$C_3^2:C_4$
Order:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism Group:	$F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Outer Automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Nilpotency class:	$-1$
Derived length:	$2$

The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.

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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3^{12}.C_2^5.A_4$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \)
$\operatorname{Aut}(H)$	not computed
$W$	$C_3^2:C_4$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)

Related subgroups

Centralizer:	$C_2\times C_6^4$
Normalizer:	$C_2^5:(\He_3^2:C_4)$
Complements:	$C_3^2:C_4$
Minimal over-subgroups:	$C_2\times C_6^2\times C_6^2:C_3$	$C_2\times C_2^4.C_3^4:C_3$	$C_3\times C_6\times (C_2^3\times C_6):S_3$
Maximal under-subgroups:	$C_6^4$	$C_6^4$	$C_6^4$	$C_6^4$	$C_6^4$	$C_2^2\times C_6^3$	$C_2^2\times C_6^3$	$C_2^2\times C_6^3$	$C_2^2\times C_6^3$	$C_2^2\times C_6^3$	$C_2^2\times C_6^3$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	not computed