#72270511

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12750 ⋊ C₂
Sylow	C₃ ⋊ #500168

Subgroups

	subgroup	G/H
center	C₂	#10828135
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12750	C₂
socle	C₂×C₃	#24033
prime radical	#11671071	C₂
hypercenter	#12750	D₃
hypocenter	C₃	#500168
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #12750

1956 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)(6,17)(7,22)(8,23)(10,24)(11,28)(12,13)(15,31)(16,34)(18,20)(19,35)(21,33)(25,30)(26,32)(27,29);(2,3)(6,16,19,20)(7,21)(10,15,26,30)(11,27)(17,34,35,18)(22,33)(24,31,32,25)(28,29);(4,6,7,18,23,16,33,35,9,19,21,34,8,20,22,17)(5,10,11,25,12,30,29,24,14,26,27,31,13,15,28,32);(4,5)(6,15)(7,11)(8,13)(9,14)(10,16)(12,23)(17,31)(18,32)(19,30)(20,26)(21,27)(22,28)(24,34)(25,35)(29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#500168, #72271549
maximal normal subgroup of	D₃×#500168, #72271549

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