#72341860

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72280972 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#120⋊C₃
commutator, cosocle	C₂×H(5,𝔽₂)	C₂⁴×C₃
Frattini	C₂⁴	C₂⁴×A₄
Fitting	#72280972	C₃
socle	C₂²	C₂³×#120⋊C₃
hypercenter	C₂×#41	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₂²×C₃×D₄
LCS	G ▷ C₂×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

At least 8832 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^6|c^2=k,d^2=k,g^2=k,h^2=k,c^a=ck,c^b=cf,g^b=gi,g^e=h,h^b=hj,h^e=ghk,h^g=hk,i^e=j,i^h=ik,j^e=ijk,j^g=jk

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341901
maximal normal subgroup of	#72341901

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