#72215489

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

C₃×#585 ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×#1020
commutator, cosocle, hypocenter	C₃×#997	C₂³
Fitting	C₃×#585	C₂×D₃
socle	C₂×C₃	C₂²×#1020
hypercenter	C₄	D₃×#1020
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂ ◁ C₄

3162 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 3162 conjugacy classes of subgroups — too many to draw. Only its 58 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:

2^3,3^2,2^5|a^2=j,f^2=j,g^2=j,c^a=cj,d^b=d^2,e^c=e^2,f^b=fj,f^d=g,g^b=fg,g^d=fgj,g^f=gj,h^b=i,h^d=i,h^f=hj,i^b=h,i^d=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#11715828, #72275031, #72275094
maximal normal subgroup of	D₃×#11715828, #72275031, #72275094

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