#11794315

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Fitting	C₃²×C₂²⋊C₄ ⋊ Q₈
Sylow	C₃² ⋊ #2062

	subgroup	G/H
center	C₂	#10593556
commutator, prime radical, cosocle	C₂³×C₃⋊D₃	C₂³
Frattini	C₂³	C₂×PSU(3,𝔽₂)
Fitting	C₃²×C₂²⋊C₄	Q₈
socle	C₂×C₃²	#395
hypercenter	C₂²⋊C₄	PSU(3,𝔽₂)
hypocenter	C₃²	#2062
LCS	G ▷ C₂³×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

380 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

μ_t(G) = 48

Polycyclic presentation:

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

maximal subgroup of	#72242747, #72242749
maximal normal subgroup of	#72242747, #72242749

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 28326);