#72220568

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ C₂×Q₈
Sylow	C₃⁴ ⋊ C₂×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#13049818
commutator, cosocle	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂⁴
Fitting	C₂²×C₃⁴	C₂×Q₈
socle	C₂×C₃⁴	C₂²×Q₈
prime radical	C₂²×C₃⋊D₃²	C₂²
hypercenter	C₂²	#13049818
hypocenter	C₃⁴	C₂×C₂²⋊Q₈
LCS	G ▷ C₂×C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1738 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1738 conjugacy classes of subgroups — too many to draw. Only its 116 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=f,c^2=f,b^a=be,c^a=cf,g^a=gh,g^c=h^2,g^d=g^2,g^f=g^2,h^a=gh^2,h^c=g,h^d=h^2,h^f=h^2,i^a=hi^2j,i^c=ij,i^d=hi,i^f=i^2,j^a=g^2h^2ij,j^c=ij^2,j^d=gh^2j,j^f=j^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220785, #72220834, #72222921
maximal normal subgroup of	#72220785, #72220834, #72222921

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