#11715741

order	768
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini	C₂²	C₂×#1020
commutator, prime radical, cosocle	C₂×C₂²⋊SL(2,𝔽₃)	C₂²
Fitting	C₂²×C₄×Q₈	D₃
socle	C₂⁴	C₂×S₄
hypercenter	C₂×C₄	#1020
hypocenter	C₂²⋊SL(2,𝔽₃)	Q₈
LCS	G ▷ C₂×C₂²⋊SL(2,𝔽₃) ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

458 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.

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

μ_t(G) = 192

Polycyclic presentation:

2^3,3,2^5|a^2=c,b^2=c,g^2=i,h^2=i,b^a=bc,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^g=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1090122);