#72209778

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11715690
commutator, cosocle, hypocenter	C₂²⋊(#95⋊C₃)	C₂²
Frattini	C₂³	C₂×#1020
Fitting	C₂²×#399	D₃
socle	C₂⁵	C₂×S₄
prime radical	#11709856	C₂
LCS	G ▷ C₂²⋊(#95⋊C₃)
UCS	C₁ ◁ C₂

1242 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 1242 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^7|b^2=hij,f^2=h,g^2=i,c^a=cij,c^b=c^2ij,d^b=e,d^c=e,e^b=d,e^c=de,f^a=fhj,f^b=gh,f^c=ghj,g^a=gij,g^b=fi,g^c=fgj,g^f=gj,h^b=i,h^c=i,i^b=h,i^c=hij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209784, #72267873
maximal normal subgroup of	#72209784, #72267873
maximal quotient of	#72209856, #72209870, #72267879

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