#72215002

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸×C₃×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²×#120 ⋊ C₂
Sylow	C₃² ⋊ #15258

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×C₃⋊D₃
commutator, socle, cosocle	C₂⁴×C₃²	C₂⁴
Fitting	C₂²×C₃²×#120	C₂
hypercenter	C₂²×#120	C₃⋊D₃
hypocenter	C₃²	#15258
LCS	G ▷ C₂⁴×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#120

15402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=e,d^2=h,b^a=bf,c^a=ce,d^b=dg,d^c=dh,i^a=i^2,j^a=j^2

Conjugacy classes & character table

282 conjugacy classes — view the full character table.

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