#11311⋊C₃ #11709167

order	768
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

#11311⋊C₃

Splittings

Fitting, Sylow

#11311 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	#396⋊C₃
commutator, hypocenter	#338	C₃×C₄
Frattini	C₂³×C₄	C₂×A₄
Fitting	#11311	C₃
socle	C₂	#1443⋊C₃
prime radical	C₂×#338⋊C₃	C₂
cosocle	C₂×#338	C₂×C₃
hypercenter	C₂×C₄	#126⋊C₃
LCS	G ▷ #338
UCS	C₁ ◁ C₄ ◁ C₂×C₄

127 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 127 conjugacy classes of subgroups — too many to draw. Only its 20 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=h,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=bcde,c^a=cdhi,d^a=cef,e^a=efi,f^a=eg,h^a=eghi,c^b=ce,d^b=df,e^b=ei,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,f^d=fi,h^d=hi

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1083548);