#72213068

order	1536
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³	C₂×#857
commutator	C₂×C₄²⋊C₃	C₄²
Frattini, socle	C₂⁵	C₂×S₄
Fitting	C₂²×C₄³	D₃
prime radical, cosocle	C₂³×C₄²⋊C₃	C₂²
hypercenter	C₂²×C₄	#857
hypocenter	C₄²⋊C₃	#95
LCS	G ▷ C₂×C₄²⋊C₃ ▷ C₄²⋊C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

1758 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 1758 conjugacy classes of subgroups — too many to draw. Only its 83 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:

2^5,3,2^4|a^2=d,b^2=c,g^2=i,h^2=j,b^a=be,f^a=f^2i,f^b=fi,f^d=fi,g^a=ghi,g^f=hi,h^f=gh,i^a=ij,i^f=j,j^f=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213103, #72218208, #72227666, …
maximal normal subgroup of	#72218208
maximal quotient of	#72213071, #72268650, #72268676

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