#72234243

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting

C₂³×#392 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#11715690
commutator, cosocle	C₂×C₂²⋊(#95⋊C₃)	C₂²
Frattini	C₂⁴	C₂×#1020
Fitting	C₂³×#392	D₃
socle	C₂⁶	C₂×S₄
hypocenter	C₂²⋊(#95⋊C₃)	D₄
LCS	G ▷ C₂×C₂²⋊(#95⋊C₃) ▷ C₂²⋊(#95⋊C₃)
UCS	C₁ ◁ C₂²

14695 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 14695 conjugacy classes of subgroups — too many to draw. Only its 42 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:

2^3,3,2^7|a^2=k,b^2=cijk,e^2=i,f^2=j,b^a=bc,d^a=dik,d^b=d^2,d^c=djk,e^a=eik,e^b=fi,e^d=fik,f^a=fjk,f^b=ej,f^d=efk,f^e=fk,g^b=hjk,g^d=h,h^b=gik,h^d=gh,i^b=j,i^d=j,j^b=i,j^d=ijk

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260440, #72261895
maximal normal subgroup of	#72260440
maximal quotient of	#72261895

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