#72278699

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#4130887 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#3532⋊C₃
commutator, cosocle, hypocenter	#17281	C₂×C₃
Frattini, socle	C₂⁴	C₂×C₂²⋊A₄
Fitting	#4130887	C₃
LCS	G ▷ #17281
UCS	C₁ ◁ C₂²

979 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 979 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 96

Polycyclic presentation:

2,3,2^8|a^2=ghij,c^2=h,d^2=j,e^2=g,f^2=g,c^a=chj,c^b=cd,d^a=dgh,d^b=cgj,d^c=dg,e^b=ef,e^c=eghj,e^d=eghi,f^b=e,f^c=fhi,f^d=fh,f^e=fg,h^b=ghj,i^b=ghi,j^b=h

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279177
maximal normal subgroup of	#72279177
maximal quotient of	#72299976

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