#72214541

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₈² ⋊ C₂²
Sylow	C₃² ⋊ #12310

Subgroups

	subgroup	G/H
center	C₂	#11778903
commutator, cosocle	C₃²×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂×D₃²
Fitting	C₃²×C₈²	C₂²
socle	C₂×C₃²	#1619
hypercenter	C₈²	D₃²
hypocenter	C₃²	#12310
LCS	G ▷ C₃²×C₄×C₈ ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

952 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 98 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:

2^8,3^2|b^2=d,d^2=f,e^2=g,g^2=h,b^a=be,c^b=cd,d^a=dg,d^c=df,e^a=egh,e^c=egh,f^a=fh,g^a=gh,g^c=gh,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219773, #72219779, #72219781
maximal normal subgroup of	#72219773, #72219779, #72219781

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