#72340994

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72333431 ⋊ C₃
derived	C₂²×H(5,𝔽₂) ⋊ C₂³×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72340980
commutator, cosocle, hypocenter	C₂²×H(5,𝔽₂)	C₂³×C₃
Frattini, socle	C₂³	C₂³×C₂²⋊A₄
Fitting	#72333431	C₃
LCS	G ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂

16168 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 16168 conjugacy classes of subgroups — too many to draw. Only its 181 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|g^2=k,h^2=k,e^c=eij,e^d=f,f^c=fi,f^d=ef,g^a=gij,g^b=gj,g^c=gi,g^d=gh,g^e=gk,g^f=gk,h^a=hj,h^b=hi,h^c=hij,h^d=g,h^e=hk,h^g=hk,i^d=j,j^d=ij

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341062
maximal normal subgroup of	#72341062

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