#11772391

order	1152
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #1381

Subgroups

	subgroup	G/H
center	C₂²	#63746
commutator	C₃⋊(C₃⋊C₈)	C₄²
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂²×C₃²	#107
prime radical	C₂²×C₃⋊D₃	Q₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1381
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^7,3^2|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=f,f^2=g,b^a=bc,c^a=cf,e^a=efg,f^a=fg,h^a=hi,i^a=hi^2,h^b=i^2,i^b=h,h^c=h^2,i^c=i^2,h^d=h^2,i^d=i^2,h^e=h^2,i^e=i^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 6402);