#72255013

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#7464 ⋊ C₂
Sylow	C₃ ⋊ #94614

Subgroups

	subgroup	G/H
center	C₂	#10682642
commutator	C₃×C₃₂	C₂×C₈
Frattini	#1397	C₂×D₃
Fitting	C₃×#7464	C₂
socle	C₂×C₃	#7385
prime radical	C₂×D₉₆	C₄
cosocle	C₃×#1397	C₂²
hypercenter	#7464	D₃
hypocenter	C₃	#94614
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397 ◁ #7464

156 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|a^2=c,b^2=fg,c^2=efhi,d^2=fgh,e^2=gh,f^2=h,g^2=hi,h^2=i,b^a=bcdfghi,c^b=ce,d^a=dh,d^b=dhi,d^c=di,e^a=ei,e^b=eg,f^a=fi,f^b=fhi,g^b=gh,h^b=hi,j^a=j^2,j^b=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255016, #72255030, D₃×#94614
maximal normal subgroup of	#72255016, #72255030, D₃×#94614

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