#72219862

order	2592
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Sylow

H(5,𝔽₂) ⋊ C₃×3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃×A₄²
commutator, cosocle	C₃×H(5,𝔽₂)	C₃³
Fitting	C₃²×H(5,𝔽₂)	C₃²
prime radical	C₃²×#997	C₃
hypercenter	C₂×C₃²	A₄²
hypocenter	H(5,𝔽₂)	C₃×3₋¹⁺²
LCS	G ▷ C₃×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃²

330 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

3^4,2^5|c^3=d,e^2=i,f^2=i,c^a=cd,e^b=f,e^c=f,f^b=efi,f^c=efi,f^e=fi,g^b=h,g^c=efg,g^f=gi,h^b=gh,h^c=ehi,h^e=hi

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219864, C₂×#72219862, #72245011, …
maximal normal subgroup of	#72219864, C₂×#72219862, #72245011, …
maximal quotient of	C₂×#72219862, #72245041, #72276184

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);