#72221981

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14554344 ⋊ C₂²
derived	#10624761 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#10624761	C₂²×C₃
Frattini	C₃³	C₃×D₃×C₃⋊D₃
Fitting	#14554344	C₂²
socle	C₃	C₃×#11757731
LCS	G ▷ #10624761
UCS	C₁

1814 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 1814 conjugacy classes of subgroups — too many to draw. Only its 63 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:

2^2,3^7|d^b=d^2,e^a=e^2,e^c=ei,e^d=eg,f^b=f^2,f^e=fgh^2,g^a=g^2,g^b=g^2i,g^d=gi,g^f=gi^2,h^a=h^2,h^b=h^2,h^f=hi,i^a=i^2

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222000, #72253859
maximal normal subgroup of	#72253859

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