#72220324

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₂³×C₃⁴ ⋊ C₂×C₄
Sylow	C₃⁴ ⋊ #386

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×#10620198
commutator, socle	C₂²×C₃⁴	C₂²×C₄
Fitting	C₂³×C₃⁴	C₂×C₄
prime radical	C₂×#13049703	C₂
cosocle	C₂²×C₃⋊(C₃⋊(C₃⋊D₃))	C₂³
hypercenter	C₂³	#10620198
hypocenter	C₃⁴	#386
LCS	G ▷ C₂²×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂² ◁ C₂³

5668 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 5668 conjugacy classes of subgroups — too many to draw. Only its 81 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:

2^6,3^4|a^2=d,b^a=be,c^b=cf,g^a=gi,g^c=g^2,g^d=g^2,h^a=hj,h^c=g^2h,h^d=h^2,i^a=gi^2,i^c=i^2,i^d=i^2,j^a=hj^2,j^c=i^2j,j^d=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

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