#72220144

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₉²×C₂²⋊C₄ ⋊ C₂²
Sylow	C₉² ⋊ B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72220023
commutator, cosocle	C₂³×C₉²	C₂³
Frattini	C₂³×C₃²	C₂×D₃²
Fitting	C₉²×C₂²⋊C₄	C₂²
socle	C₂×C₃²	#63923
hypercenter	C₂²⋊C₄	D₉²
hypocenter	C₉²	B(4,𝔽₂)
LCS	G ▷ C₂³×C₉² ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

1346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1346 conjugacy classes of subgroups — too many to draw. Only its 103 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:

2^6,3^4|a^2=f,g^3=i,h^3=j,c^a=ce,c^b=cd,d^a=df,e^b=ef,g^a=g^2i^2,g^b=g^2i^2,g^c=g^2i^2,h^a=g^2h,h^b=h^2j^2,h^c=h^2j^2,i^a=i^2,i^b=i^2,i^c=i^2,j^a=i^2j,j^b=j^2,j^c=j^2

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220147, #72220169
maximal normal subgroup of	#72220147, #72220169

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