#72218415

order	2304
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₂⁸ ⋊ C₃²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁸	C₃²
LCS	G ▷ C₂⁸
UCS	C₁

47257 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 47257 conjugacy classes of subgroups — too many to draw. Only its 41 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:

3^2,2^8|c^a=ch,c^b=cd,d^a=dgh,d^b=c,e^a=cef,f^a=ceh,f^b=df,g^a=cd,g^b=gh,h^a=c,h^b=g,i^a=j,i^b=ij,j^a=ij,j^b=i

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218837, #72226400, #72226446
maximal normal subgroup of	#72218837, #72226400, #72226446
maximal quotient of	#72272663

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