#72271773

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#10953 ⋊ C₂
Sylow	C₃ ⋊ #375718

Subgroups

	subgroup	G/H
center	C₂	#10720466
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₃×#10953	C₂
socle	C₂×C₃	#14169
prime radical	C₂×#5805	C₂²
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	#10953	D₃
hypocenter	C₃	#375718
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10953

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

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,c^2=h,g^2=i,b^a=bgh,c^a=ci,c^b=cgh,d^a=dgh,d^b=df,d^c=dh,e^a=ei,e^c=ei,f^a=fi,g^b=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, 3, F.8);;
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.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, 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, 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);