#72279199

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#158869 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#15258⋊C₃
commutator, cosocle, hypocenter	C₂²×#95	C₂²×C₃
Frattini, socle	C₂⁵	C₂²×A₄
Fitting	#158869	C₃
prime radical	#11709839	C₂
LCS	G ▷ C₂²×#95
UCS	C₁ ◁ C₂

2265 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 2265 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) = 48

Polycyclic presentation:

2^2,3,2^7|a^2=j,b^2=j,d^2=h,e^2=hi,b^a=bj,c^a=chij,c^b=chj,d^a=dhj,d^b=df,d^c=de,e^a=ehij,e^b=efg,e^c=dhij,e^d=ej,f^c=g,g^c=fg,h^c=ij,i^c=hij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299775
maximal quotient of	#72299978

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