#72221920

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72221882 ⋊ C₂
derived	C₃³×C₉ ⋊ C₂×C₃³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂×C₃³
Frattini	C₃³	C₃³×C₃⋊D₃
Fitting	#72221882	C₂
socle	C₃	C₃×#13091503
LCS	G ▷ C₃³×C₉
UCS	C₁

8283 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 8283 conjugacy classes of subgroups — too many to draw. Only its 251 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:

2,3^8|f^3=i,e^a=e^2,e^c=eh,e^d=eg,f^a=f^2i^2,f^b=fi,f^c=fg^2,f^d=fg^2,g^a=g^2,g^c=gi^2,g^d=gi^2,h^a=h^2,h^d=hi^2,i^a=i^2

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222003, #72243955, C₂×#72221920
maximal normal subgroup of	#72222003, #72243955, C₂×#72221920
maximal quotient of	C₂×#72221920

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