#13180176

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

C₃²×Q₈ ⋊ Dic₆

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄⋊PSU(3,𝔽₂)
commutator, prime radical, cosocle	C₃⋊D₃×SL(2,𝔽₃)	C₂²
Fitting	C₃²×Q₈	Dic₆
socle	C₂×C₃²	A₄⋊Q₈
hypocenter	C₃²×SL(2,𝔽₃)	Q₈
LCS	G ▷ C₃⋊D₃×SL(2,𝔽₃) ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

154 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

μ_t(G) = 72

Polycyclic presentation:

2^3,3^3,2^3|a^2=c,b^2=c,g^2=i,h^2=i,b^a=bc,d^a=d^2,e^a=f,f^a=e^2,g^a=gh,h^a=hi,e^b=ef,f^b=ef^2,e^c=e^2,f^c=f^2,g^d=h,h^d=ghi,h^g=hi

maximal subgroup of	GL(2,𝔽₃)×PSU(3,𝔽₂), #72244565, #72244569
maximal normal subgroup of	GL(2,𝔽₃)×PSU(3,𝔽₂), #72244565, #72244569

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46239);