#72220003

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄×#6900 ⋊ C₂²
Sylow	#6900 ⋊ C₂×C₂²⋊C₄

Subgroups

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

1692 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1692 conjugacy classes of subgroups — too many to draw. Only its 172 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|b^2=e,g^3=i,h^3=j,b^a=bf,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

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220091
maximal normal subgroup of	#72220091
maximal quotient of	#72220005, #72220151

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, 2, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*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);