#72255419

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1369 ⋊ C₂
Sylow	C₃² ⋊ #12068

Subgroups

	subgroup	G/H
center	C₄	#10592568
commutator	C₃²×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×#1369	C₂
socle	C₂×C₃²	#1580
prime radical	#10592603	C₄
cosocle	C₃²×C₄×C₈	C₂³
hypercenter	#1369	C₃⋊D₃
hypocenter	C₃²	#12068
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

612 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Polycyclic presentation:

2^8,3^2|a^2=d,b^2=defg,c^2=eg,d^2=f,e^2=g,f^2=h,g^2=h,b^a=be,c^a=ceg,c^b=cgh,d^b=dh,d^c=dh,e^a=eg,e^b=eh,g^a=gh,i^b=i^2,j^b=j^2

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#12068, #72255461, #72255470
maximal normal subgroup of	C₃⋊D₃×#12068, #72255461, #72255470
maximal quotient of	#72269777

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