#72221972

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

C₃²⋊H(3,𝔽₃) ⋊ SD₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	#66472⋊C₃	C₂²
Frattini, socle	C₃²	#89092
Fitting, hypocenter	C₃²⋊H(3,𝔽₃)	SD₁₆
prime radical	#14517930	C₂
LCS	G ▷ #66472⋊C₃ ▷ #92301 ▷ C₃²⋊H(3,𝔽₃)
UCS	C₁

202 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 14 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|b^2=d,c^2=d,b^a=bc,c^a=cd,c^b=cd,e^a=e^2,f^a=f^2,f^b=fg,f^c=g^2,f^d=f^2,f^e=fh,g^b=fg^2,g^c=f,g^d=g^2,g^e=ghi^2,h^b=h^2i^2,h^c=h^2i,h^d=h^2,i^a=h^2i^2,i^b=h^2i,i^c=hi,i^d=i^2

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221979, #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, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.7^2);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8*F.9^2);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);