#72235728

order	1536
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#154067 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#13829⋊C₃
commutator, Fitting, cosocle, hypocenter	#154067	C₃
Frattini, socle	C₂⁵	C₂²⋊A₄
LCS	G ▷ #154067
UCS	C₁ ◁ C₂

873 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 48

Polycyclic presentation:

3,2^9|b^2=f,c^2=g,d^2=ghi,e^2=fghij,b^a=cfh,c^a=bch,c^b=ch,d^a=egi,d^b=di,d^c=dj,e^a=deh,e^b=ehj,e^c=ehij,e^d=eh,f^a=g,g^a=fgh,i^a=hij,j^a=i

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299680, #72299688
maximal normal subgroup of	#72299680, #72299688
maximal quotient of	#72235730, #72240029, #72299953

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