#72219903

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

#6900 ⋊ C₂×SD₁₆

Subgroups

	subgroup	G/H
center	C₂	#72213202
commutator, cosocle	C₄×C₉²	C₂³×C₃
Frattini	C₃²×C₄	C₂×C₃×D₃²
Fitting	D₄×#6900	C₂²
socle	C₂×C₃²	C₃×#3811
prime radical	D₄×#92314	C₂
hypercenter	D₄	#92265⋊C₂
hypocenter	C₉²	C₂×C₃×SD₁₆
LCS	G ▷ C₄×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

1074 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 106 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:

2^3,3,2^2,3^4|a^2=e,e^2=f,g^3=i,h^3=j,b^a=bef,e^b=ef,g^a=g^2i^2,g^c=g^2i^2,g^d=gi^2,h^a=g^2h,h^c=h^2j^2,h^d=hj^2,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219937
maximal normal subgroup of	#72219937

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);