#11772546

order	1152
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #1389

Subgroups

	subgroup	G/H
center	C₂	#10593166
commutator	#622	C₄²
Frattini, hypercenter	C₂³	AΓL(1,𝔽₉)
Fitting, prime radical	C₂³×C₃²	SD₁₆
socle	C₂×C₃²	#340
cosocle	C₂²×#622	C₂²
hypocenter	C₃²	#1389
LCS	G ▷ #622 ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³

166 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 44 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|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=cf,e^a=ef,f^a=fg,h^a=hi^2,i^a=h^2i^2,c^b=cg,h^b=h^2i,i^b=h,h^c=h^2i^2,i^c=h^2i,h^d=h^2,i^d=i^2,h^e=h^2i^2,i^e=h^2i,h^f=h^2,i^f=i^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 6557);