#72213893

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×AGL(1,ℤ/8) ⋊ C₂²
Sylow	C₃² ⋊ C₂×#3585

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₃²×D₄
commutator, cosocle	C₃²×C₄	C₂⁶
Frattini	C₄	C₂⁴×D₃²
Fitting	C₂×C₃²×AGL(1,ℤ/8)	C₂²
socle	C₂×C₃²	C₂⁴×D₄
hypercenter	C₂×AGL(1,ℤ/8)	D₃²
hypocenter	C₃²	C₂×#3585
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

23136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219759
maximal normal subgroup of	#72215117, #72219759
maximal quotient of	#72228219
holomorph of	D₃×C₈, ΓL(1,𝔽₂₅)

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