#11779651

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #1637

Subgroups

	subgroup	G/H
center	C₂²	#64241
commutator, cosocle	C₄×C₃⋊Dic₃	C₂³
Frattini	C₂×C₄	C₂×GO⁺(4,𝔽₂)
Fitting	C₃²×C₄⋊C₄	D₄
socle	C₂²×C₃²	#120
prime radical	C₂×C₄⋊(C₃⋊D₃)	D₄
hypercenter	C₄⋊C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	#1637
LCS	G ▷ C₄×C₃⋊Dic₃ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊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) = 192

Polycyclic presentation:

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

54 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 13662);