#72227587

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#72219811 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂⁴	C₂²⋊(C₂²⋊A₄)
commutator, Fitting, cosocle, hypocenter	#72219811	C₃
LCS	G ▷ #72219811
UCS	C₁ ◁ C₂⁴

40861 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 40861 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 64

Polycyclic presentation:

3,2^10|d^2=i,e^2=i,f^2=hijk,g^2=hijk,b^a=c,c^a=bc,d^a=de,d^b=dk,d^c=dh,e^a=d,e^b=eh,e^c=ehk,e^d=ei,f^a=g,f^b=fhk,f^c=fk,f^d=fij,f^e=fhik,g^a=fghijk,g^b=gh,g^c=ghk,g^d=ghik,g^e=ghjk,g^f=ghijk

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227611, #72227701
maximal subgroup of	#72227611, #72227701
maximal normal subgroup of	#72227611, #72227701

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", "k");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 7, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.9*F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);