#72226204

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₁₆

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#122220 ⋊ C₂
Sylow	C₃ ⋊ #72214020

Subgroups

	subgroup	G/H
center	C₂²	#10695025
commutator	C₃×C₈²	C₂²×C₄
Frattini	C₂×C₈²	C₂²×D₃
Fitting	C₃×#122220	C₂
socle	C₂²×C₃	#9837
prime radical	#72225817	C₂
cosocle	C₂×C₃×C₈²	C₂³
hypercenter	#122220	D₃
hypocenter	C₃	#72214020
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122220

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(20,21,22,25,23,28,29,26,24,27,31,33,32,35,30,34);(2,3)(5,9)(6,14)(7,17)(10,11)(12,19)(13,16)(15,18)(21,26)(22,30)(23,32)(25,28)(27,34)(29,31)(33,35);(2,3)(4,5,6,10,8,13,14,12)(7,16,15,11,18,9,17,19)(21,25,27,33)(22,29)(23,32)(26,28,34,35)(30,31)

Polycyclic presentation:

2^10,3|a^2=d,c^2=f,d^2=i,e^2=g,f^2=h,g^2=i,h^2=j,b^a=be,c^a=cf,c^b=cfh,d^b=di,d^c=dj,e^a=eg,e^b=egi,f^a=fh,f^b=fhj,g^a=gi,g^b=gi,h^a=hj,h^b=hj,k^a=k^2,k^b=k^2

Conjugacy classes & character table

204 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, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> SetConjugate(coll, 11, 2, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);