#72220148

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

derived

C₃×C₂²⋊(C₂²⋊C₉) ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₂²⋊(C₂²⋊C₉)	C₂×C₃
Frattini	C₃	C₃×C₃⋊#1020
Fitting	C₂⁴×C₃³	D₃
socle	C₂⁴×C₃²	C₃×D₃
LCS	G ▷ C₃×C₂²⋊(C₂²⋊C₉)
UCS	C₁

851 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 851 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^4,2^4|c^3=e,c^a=c^2e^2,c^b=ce^2,d^a=d^2,e^a=e^2,f^a=g,f^c=g,g^a=f,g^c=fg,h^a=fi,h^c=hi,i^a=gh,i^c=h

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220160, C₂×#72220148
maximal normal subgroup of	C₂×#72220148
maximal quotient of	C₂×#72220148, #72252942

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