#72271688

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13417 ⋊ C₂
Sylow	C₃ ⋊ #500164

Subgroups

	subgroup	G/H
center	C₂	#10828132
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13417	C₂
socle	C₂×C₃	#24033
prime radical	#11670499	C₂
hypercenter	#13417	D₃
hypocenter	C₃	#500164
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #13417

2080 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:

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=ghi,e^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ce,d^b=defhi,d^c=dh,e^a=ei,e^b=eh,e^c=ehi,e^d=ehi,f^a=fi,g^b=gi,h^b=hi,h^c=hi,h^d=hi,j^d=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271714, #72271782
maximal normal subgroup of	#72271714, #72271782

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