#72270264

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12750 ⋊ C₂
Sylow	C₃ ⋊ #495739

Subgroups

	subgroup	G/H
center	C₂	#10800825
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12750	C₂
socle	C₂×C₃	#21452
prime radical	#10949153	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#12750	D₃
hypocenter	C₃	#495739
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12750

2106 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)(6,16,18,19)(7,21)(10,24,14,26)(11,28)(15,32,33,17)(20,34)(23,31,30,25)(27,35);(6,15)(7,20)(8,22)(10,23)(11,27)(13,29)(14,30)(16,32)(17,19)(18,33)(21,34)(24,31)(25,26)(28,35);(2,3)(4,6,7,17,22,16,34,33,9,18,21,32,8,19,20,15)(5,10,11,25,29,24,35,30,12,14,28,31,13,26,27,23);(2,3)(4,5)(6,14,18,10)(7,11)(8,13)(9,12)(15,30,33,23)(16,26,19,24)(17,31,32,25)(20,27)(21,28)(22,29)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495739
maximal normal subgroup of	D₃×#495739

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