#72268698

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Sylow

#6900 ⋊ C₄⋊C₄

Subgroups

	subgroup	G/H
center	C₂²	C₂×#92314
commutator	C₂×C₉²	C₂×C₃×C₄
Frattini, socle	C₂²×C₃²	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₄×#6900	C₂
prime radical	C₂²×C₃³	C₂×C₃⋊D₃
cosocle	C₂²×C₉²	C₂²×C₃
hypercenter	C₂×C₄	#92314
hypocenter	C₉²	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1296

Polycyclic presentation:

2^2,3,2^2,3^4|a^2=d,b^2=e,f^3=h,g^3=i,b^a=be,f^b=f^2h^2,f^c=fh,g^b=g^2i^2,g^c=gi,h^b=h^2,i^b=i^2

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄⋊C₄×#92314
maximal normal subgroup of	C₄⋊C₄×#92314
maximal quotient of	#72268715

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