#72214760

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#3484 ⋊ C₂
Sylow	C₃² ⋊ #23900

Subgroups

	subgroup	G/H
center	C₂²	C₂×#64162
commutator, cosocle	C₂⁴×C₃²	C₂⁴
Frattini	C₂⁴	C₂³×C₃⋊D₃
Fitting	C₃²×#3484	C₂
socle	C₂²×C₃²	C₂×#120
hypercenter	#3484	C₃⋊D₃
hypocenter	C₃²	#23900
LCS	G ▷ C₂⁴×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #3484

10560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|c^2=e,d^2=h,c^a=ce,c^b=cef,d^a=dh,d^b=dg,f^d=fh,g^c=gh,i^b=i^2,j^b=j^2

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252331
maximal normal subgroup of	#72252331

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.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, 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);