#72271774

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#10950 ⋊ C₂
Sylow	C₃ ⋊ #375718

Subgroups

	subgroup	G/H
center	C₂	#10720466
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₃×#10950	C₂
socle	C₂×C₃	#14169
prime radical	C₂×#5830	C₂²
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	#10950	D₃
hypocenter	C₃	#375718
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10950

2234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,d^2=i,g^2=i,b^a=bgh,c^a=cgh,c^b=cf,d^a=dghi,d^b=dfghi,d^c=dhi,e^a=ei,e^d=ei,f^a=fi,g^b=gi,g^c=gi,g^d=gi,h^b=hi,j^b=j^2,j^c=j^2

Conjugacy classes & character table

135 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);