#72220680

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ D₄
Sylow	C₃⁴ ⋊ #120

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#10620203
commutator, cosocle	C₂×C₃²×C₃⋊D₃	C₂³
Fitting	C₂²×C₃⁴	D₄
socle	C₂×C₃⁴	C₂×D₄
hypercenter	C₂²	#10620203
hypocenter	C₃⁴	#120
LCS	G ▷ C₂×C₃²×C₃⋊D₃ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1262 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 72 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:

2^5,3^4|a^2=e,b^a=bd,c^a=ce,f^a=fg,f^c=f^2g^2,f^e=f^2,g^a=fg^2,g^e=g^2,h^a=f^2g^2h^2,h^c=h^2,h^e=f^2gh,i^a=g^2i^2,i^c=hi,i^e=f^2g^2i

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220706
maximal normal subgroup of	#72220706

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