#72276144

order	7776
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

H(5,𝔽₂) ⋊ #6875

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#13048169
commutator	C₉×H(5,𝔽₂)	C₃×C₉
Frattini	C₂×C₃×C₉	C₃×C₂²⋊A₄
Fitting	H(5,𝔽₂)×C₉⋊C₉	C₃
prime radical	C₃²×H(5,𝔽₂)	3₋¹⁺²
cosocle	C₃×C₉×H(5,𝔽₂)	C₃²
hypercenter	C₂×C₉⋊C₉	C₂²⋊A₄
hypocenter	H(5,𝔽₂)	#6875
LCS	G ▷ C₉×H(5,𝔽₂) ▷ C₃×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₉ ◁ C₂×C₉⋊C₉

628 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 216

Polycyclic presentation:

3^5,2^5|a^3=c,b^3=d,d^3=e^2,f^2=j,g^2=j,b^a=bd,c^b=ce,d^a=de^2,f^b=g,g^b=fgj,g^f=gj,h^b=fhi,h^g=hj,i^b=ghj,i^f=ij

Conjugacy classes & character table

235 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72276206
maximal normal subgroup of	#72276206

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", "j");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5^2);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5^2);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);