#72230367

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#7307 ⋊ C₂
Sylow	C₃ ⋊ #144366

Subgroups

	subgroup	G/H
center	C₂²	#68965
commutator, cosocle	C₂³×C₃×D₄	C₂³
Frattini	C₂³×D₄	C₂²×D₃
Fitting	C₃×#7307	C₂
socle	C₂²×C₃	#2021
hypercenter	#7307	D₃
hypocenter	C₃	#144366
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #7307

4166 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 4166 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^9,3|b^a=bf,c^a=ce,c^b=cd,d^a=dh,e^b=eg,f^c=fghi,f^d=fi,h^b=hi,j^a=j^2,j^b=j^2,j^c=j^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#144366, #72231829
maximal normal subgroup of	D₃×#144366, #72231829
maximal quotient of	#72270635

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