#12772⋊C₃ #11709183

order	768
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

#12772⋊C₃

Splittings

Fitting, Sylow

#12772 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	B(4,𝔽₂)⋊C₃
commutator, hypocenter	H(3,𝔽₄)	C₃×C₄
Frattini	C₂×#41	C₂×A₄
Fitting	#12772	C₃
socle	C₂	#1790⋊C₃
prime radical	#3529⋊C₃	C₂
cosocle	#3529	C₂×C₃
hypercenter	C₂×C₄	#120⋊C₃
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₄ ◁ C₂×C₄

180 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

Polycyclic presentation:

3,2^8|b^2=g,e^2=i,f^2=i,b^a=bcdfhi,c^a=cdhi,d^a=ci,e^a=efi,f^a=ei,g^a=efg,c^b=ce,d^b=df,e^b=eh,h^b=hi,e^c=ei,f^c=fhi,g^c=ghi,e^d=eh,f^d=fi,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1083564);