#72258773

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting

#10574840 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₄×S₄
commutator, cosocle	C₂×#41⋊C₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×S₄
Fitting	#10574840	D₃
hypercenter	#3004	S₄
hypocenter	SL(2,𝔽₃)	C₂×D₄²
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #3004

At least 7075 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,3,2^3|a^2=f,e^2=k,g^2=k,i^2=k,j^2=k,b^a=bf,d^a=dk,e^c=eg,g^c=gk,g^e=gk,h^a=h^2,h^c=h^2,i^a=jk,i^c=jk,i^h=j,j^a=ik,j^c=ik,j^h=ijk,j^i=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

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