#13164073

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

	subgroup	G/H
center, hypercenter	C₂	A₄⋊((C₂²⋊C₉)⋊C₂)
commutator, cosocle, hypocenter	SL(2,𝔽₃)⋊(C₂²⋊C₉)	C₂
Frattini, socle	C₂×C₃	A₄⋊S₄
Fitting	C₃×H(5,𝔽₂)	C₃⋊D₃
prime radical	C₃×#994	D₃
LCS	G ▷ SL(2,𝔽₃)⋊(C₂²⋊C₉)
UCS	C₁ ◁ C₂

196 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.

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

μ_t(G) = 432

Polycyclic presentation:

2,3^3,2^5|a^2=i,c^3=d,e^2=i,f^2=i,g^2=i,h^2=i,b^a=b^2,c^a=c^2d^2,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,h^a=hi,g^b=h,h^b=ghi,e^c=f,f^c=efi,f^e=fi,h^g=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 30136);