#72258010

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72237302 ⋊ C₃
derived	C₂²×H(3,𝔽₄) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×H(3,𝔽₄)	C₂²×C₃
Frattini, socle	C₂²	C₂²×C₂²⋊(C₂²⋊A₄)
Fitting	#72237302	C₃
LCS	G ▷ C₂²×H(3,𝔽₄)
UCS	C₁

At least 39009 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^8|d^c=e,e^c=de,f^a=fjk,f^c=g,g^a=gk,g^c=fg,h^b=hjk,h^c=i,h^d=hj,h^e=hk,i^b=ik,i^c=hi,i^d=ik,i^e=ijk,j^c=jk,k^c=j

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258063, #72262224
maximal normal subgroup of	#72258063, #72262224

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