#72221995

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

C₃²⋊H(3,𝔽₃) ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	#92301	C₂³
Frattini, socle	C₃²	D₃×GO⁺(4,𝔽₂)
Fitting, hypocenter	C₃²⋊H(3,𝔽₃)	C₂×D₄
LCS	G ▷ #92301 ▷ C₃²⋊H(3,𝔽₃)
UCS	C₁

634 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 29 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:

2^4,3^5|c^2=d,c^b=cd,e^a=e^2,e^b=e^2,f^a=f^2,f^b=f^2g^2,f^c=f^2g^2,f^d=f^2,f^e=fh,g^a=g^2,g^c=f^2g,g^d=g^2,g^e=gh^2i,h^b=i,h^c=i^2,h^d=h^2,i^b=h,i^c=h,i^d=i^2

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222012
maximal normal subgroup of	#72222012

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