#72226128

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#11404 ⋊ C₂
Sylow	C₃ ⋊ #483961

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4711
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#11404	C₂
socle	C₂²×C₃	C₂×#442
prime radical	#10950883	C₂
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#11404	D₃
hypocenter	C₃	#483961
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ #11404

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Polycyclic presentation:

2^9,3|b^2=e,c^2=fhi,d^2=hi,e^2=h,f^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cfgh,c^b=cgi,d^a=dh,d^b=di,e^a=eg,e^c=ei,f^a=fg,f^b=fi,g^a=gi,h^a=hi,j^a=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226131
maximal normal subgroup of	#72226131

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