#72214792

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ C₂×#2165

Subgroups

	subgroup	G/H
center	C₄	#10594969
commutator	C₂×C₃²×C₄	C₂³×C₄
Frattini	C₂×C₈	C₂²×D₃²
Fitting	C₃²×C₄×M₁₆	C₂²
socle	C₂×C₃²	C₂×#432
prime radical	C₂³×C₃⋊D₃	M₁₆
cosocle	C₂×C₃²×C₈	C₂⁴
hypercenter	C₄×M₁₆	D₃²
hypocenter	C₃²	C₂×#2165
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ #97 ◁ C₄×M₁₆

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,3^2|a^2=e,d^2=fgh,e^2=g,g^2=h,b^a=bf,d^a=dfg,d^b=dh,e^b=eh,f^a=fh,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219771, #72219797, #72219799
maximal normal subgroup of	#72219771, #72219797, #72219799

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