#11794002

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #2053

Subgroups

	subgroup	G/H
center	C₂	#10593456
commutator, cosocle	C₂²×C₃⋊Dic₃	C₂³
Frattini	C₂³	C₂×GO⁺(4,𝔽₂)
Fitting	C₃²×C₂²⋊C₄	D₄
socle	C₂×C₃²	#394
prime radical	#10596532	C₂
hypercenter	C₂²⋊C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	#2053
LCS	G ▷ C₂²×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

422 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 47 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|c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=hi,i^a=i^2,c^b=cf,d^b=dg,f^b=fg,i^b=hi^2,h^c=hi,i^c=hi^2,h^d=h^2,i^d=i^2,h^e=h^2,i^e=i^2,h^f=h^2,i^f=i^2

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 28013);