#72215828

order	2304
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#62817 ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#62817	C₃²
Frattini, socle	C₂²	#10596782
LCS	G ▷ #62817
UCS	C₁

734 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 17 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:

3^2,2^8|c^b=cd,d^b=c,e^a=eh,e^b=ef,e^c=ej,f^a=fgh,f^b=e,f^c=fj,f^d=fj,g^a=ef,g^b=gh,g^d=gi,h^a=e,h^b=g,h^c=hi,i^a=j,j^a=ij

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₂×#72215868, C₂×#72215896, #72247788, …
maximal subgroup of	#72215845, #72215854, #72215868, …
maximal normal subgroup of	#72215845, #72215854, #72215868, …
maximal quotient of	C₂×#72215828, C₃×#72215828

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