#11786808

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #1861

Subgroups

	subgroup	G/H
center	C₄	#64241
commutator	C₂×C₃⋊Dic₃	C₂²×C₄
Frattini	C₂×C₄	C₂×GO⁺(4,𝔽₂)
Fitting	C₂×C₃²×C₈	D₄
socle	C₂×C₃²	#385
prime radical	#64306	C₂²
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	C₂×C₈	GO⁺(4,𝔽₂)
hypocenter	C₃²	#1861
LCS	G ▷ C₂×C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

454 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 48

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 20819);