#72214404

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14546560 ⋊ C₂²
derived	C₂₇² ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂₇²	C₂²×C₃
Frattini	C₉²	C₃×D₃²
Fitting	#14546560	C₂²
socle	C₃²	C₃×D₉²
LCS	G ▷ C₂₇²
UCS	C₁

332 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 43 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:

2^2,3^7|d^3=f,e^3=g,f^3=h,g^3=i,d^a=d^2f^2h^2,d^b=d^2f^2h^2,d^c=dh^2,e^a=d^2e,e^b=e^2g^2i^2,e^c=ei^2,f^a=f^2h^2,f^b=f^2h^2,g^a=f^2g,g^b=g^2i^2,h^a=h^2,h^b=h^2,i^a=h^2i,i^b=i^2

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214444
maximal normal subgroup of	#72214444

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