#11800114

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #2242

Subgroups

	subgroup	G/H
center	C₂	#10596402
commutator, cosocle	#3787	C₂³
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₃²×C₈	SD₁₆
socle	C₂×C₃²	#486
prime radical	#64225	C₂²
hypercenter	C₈	AΓL(1,𝔽₉)
hypocenter	C₃²	#2242
LCS	G ▷ #3787 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

172 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^7,3^2|b^2=eg,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,h^a=hi,i^a=i^2,h^b=hi^2,i^b=h^2,h^c=h^2i,i^c=h,h^d=h^2i^2,i^d=h^2i,h^e=h^2i^2,i^e=h^2i,h^f=h^2,i^f=i^2

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 34125);