#72214973

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×#104 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#2956

Subgroups

	subgroup	G/H
center	C₄	C₂×D₃²×D₄
commutator	C₃²×C₄	C₂⁴×C₄
Frattini	C₂×C₄	C₂³×D₃²
Fitting	C₂×C₃²×#104	C₂²
socle	C₂×C₃²	C₂²×C₄×D₄
prime radical	#11918749	C₂
cosocle	C₂×C₃²×C₄	C₂⁵
hypercenter	C₂×#104	D₃²
hypocenter	C₃²	C₂×#2956
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

5776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|c^2=f,g^2=h,b^a=bg,c^a=cg,c^b=cgh,d^a=dh,f^a=fh,f^b=fh,g^a=gh,g^b=gh,i^a=i^2,i^e=i^2,j^a=i^2j,j^e=j^2

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

#72215117

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