#72210232

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#10624572 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#10624572	C₂²
Frattini, socle	C₃³	#1113
LCS	G ▷ #10624572
UCS	C₁

187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3^6|c^3=f,d^3=f^2gh^2,e^3=f^2h^2,c^a=c^2f^2,d^b=d^2fg^2h,d^c=dfg^2,e^a=d^2e^2f^2gh,e^b=ce^2gh^2,e^c=ef^2gh,e^d=ef^2g^2,f^a=f^2,g^a=g^2,g^b=f^2g^2,h^a=f^2gh,h^b=h^2

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210236
maximal normal subgroup of	#72210236
maximal quotient of	#72210233

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