#72340592

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#7700073 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#63011⋊C₃
commutator, cosocle, hypocenter	#2679	C₂²×C₃
Frattini, socle	C₂³	C₂²×C₂²⋊A₄
Fitting	#7700073	C₃
prime radical	#59816⋊C₃	C₂
LCS	G ▷ #2679
UCS	C₁ ◁ C₂

2318 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 2318 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 24

Polycyclic presentation:

2^2,3,2^7|a^2=i,b^2=i,d^2=h,e^2=hj,b^a=bi,d^a=dhi,d^b=dj,d^c=ehi,e^a=ehij,e^b=ehi,e^c=dei,e^d=ei,f^a=fhi,f^c=fg,f^d=fhj,f^e=fhi,g^a=gj,g^c=f,g^d=gj,g^e=ghj,h^c=hj,j^c=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340595
maximal normal subgroup of	#72340595

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