#72226300

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #483600

Subgroups

	subgroup	G/H
center	C₄	C₂×#5152
commutator	C₃×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₃×#33463	C₂
socle	C₂×C₃	C₂×#1580
prime radical	C₃×#3414	C₄
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	#33463	D₃
hypocenter	C₃	#483600
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #33463

934 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)(5,12,14,15)(10,13,25,26)(11,28,30,29)(16,32,33,24);(4,6,18,35,8,21,22,27,9,19,31,34,23,17,7,20)(5,11,10,24,15,29,26,33,14,30,25,32,12,28,13,16);(2,3)(5,10,12,13,14,25,15,26)(6,17,19,21)(7,22)(11,16,29,24,30,33,28,32)(18,31)(20,35,27,34);(4,5,9,14)(6,16,19,33)(7,10,22,25)(8,12,23,15)(11,27,30,20)(13,31,26,18)(17,24,21,32)(28,34,29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226305
maximal normal subgroup of	#72226305

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