#72209807

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11715758
commutator, cosocle, hypocenter	#2679⋊C₃	C₂²
Frattini, socle	C₂³	C₂×#1020
Fitting	#60045	D₃
prime radical	#60045⋊C₃	C₂
LCS	G ▷ #2679⋊C₃
UCS	C₁ ◁ C₂

626 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 20 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=hj,d^2=i,e^2=ij,f^2=hi,g^2=h,c^b=c^2,d^b=ei,d^c=ei,e^b=dij,e^c=de,f^a=fij,f^b=fg,f^c=gi,f^d=fh,f^e=fij,g^a=gj,g^b=gh,g^c=fghi,g^d=ghi,g^e=ghij,g^f=ghj,h^c=hij,i^b=ij,i^c=ij,j^c=i

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209811, #72340820, #72340842, …
maximal normal subgroup of	#72209811, #72340820, #72340842, …
maximal quotient of	#72227674

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