#72219992

order	15552
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×#6900 ⋊ C₂²
Sylow	#6900 ⋊ C₂×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	C₂³×#92265⋊C₂
commutator, cosocle	C₂×C₉²	C₂⁵×C₃
Frattini, socle	C₂×C₃²	C₂³×C₃×D₃²
Fitting	C₂×D₄×#6900	C₂²
hypercenter	C₂×D₄	#92265⋊C₂
hypocenter	C₉²	C₂×C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂ ◁ C₂×D₄

10313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3,2,3^4|a^2=g,h^3=j,i^3=k,c^a=cg,d^b=dg,h^b=h^2j^2,h^e=h^2j^2,h^f=hj,i^b=h^2i,i^e=i^2k^2,i^f=ik,j^b=j^2,j^e=j^2,k^b=j^2k,k^e=k^2

Conjugacy classes & character table

318 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220092, #72220184
maximal normal subgroup of	#72220092, #72220184
holomorph of	C₄×D₉, #611

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 5, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 5, F.9^2*F.11^2);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> SetConjugate(coll, 11, 2, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 5, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);