#72213877

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃³

Aliases

No aliases for this group.

Splittings

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

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#10620211
commutator, cosocle	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂³
Fitting	C₂²×C₃⁴	Q₈
socle	C₂×C₃⁴	C₂×Q₈
prime radical	C₂²×C₃⋊(C₃⋊(C₃⋊D₃))	C₂²
hypercenter	C₂²	#10620211
hypocenter	C₃⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1157 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1157 conjugacy classes of subgroups — too many to draw. Only its 39 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=d,c^2=e,b^a=bde,c^a=ce,f^a=fg,f^c=g,f^d=f^2,f^e=f^2,g^a=fg^2,g^c=f^2,g^d=g^2,g^e=g^2,h^a=f^2g^2i,h^c=h^2i^2,h^d=h^2,h^e=h^2,i^a=f^2h^2,i^c=h^2i,i^d=i^2,i^e=i^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213898, #72219724, #72219725, …
maximal normal subgroup of	#72213898, #72219724, #72219725, …

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