#72255456

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1369 ⋊ C₂
Sylow	C₃² ⋊ #7390

Subgroups

	subgroup	G/H
center	C₂	#11771600
commutator	C₃²×C₁₆	C₂×C₈
Frattini	#346	C₂×C₃⋊D₃
Fitting	C₃²×#1369	C₂
socle	C₂×C₃²	#1333
prime radical	C₂×#63626	C₄
cosocle	C₃²×#346	C₂²
hypercenter	#1369	C₃⋊D₃
hypocenter	C₃²	#7390
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1369

384 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 288

Polycyclic presentation:

2^8,3^2|a^2=c,c^2=e,d^2=fh,e^2=gh,f^2=g,g^2=h,b^a=bd,c^b=cf,d^a=dg,d^b=dfg,d^c=dh,e^b=eg,f^a=fh,f^b=fgh,g^b=gh,i^a=i^2,i^b=i^2,j^a=j^2,j^b=j^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#7390, #72255461, #72255467
maximal normal subgroup of	C₃⋊D₃×#7390, #72255461, #72255467
maximal quotient of	#72269786

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