#72340508

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72238249 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#63011⋊C₃
commutator, cosocle	C₂×H(3,𝔽₄)	C₂³×C₃
Frattini, socle	C₂³	C₂³×C₂²⋊A₄
Fitting	#72238249	C₃
hypercenter	C₄	#63011⋊C₃
hypocenter	H(3,𝔽₄)	C₂×C₃×D₄
LCS	G ▷ C₂×H(3,𝔽₄) ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₂ ◁ C₄

27769 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 27769 conjugacy classes of subgroups — too many to draw. Only its 155 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|c^2=e,f^2=j,g^2=jk,c^b=ce,f^a=fj,f^b=fk,f^d=gj,g^a=gjk,g^b=gj,g^d=fg,h^a=hj,h^d=hi,h^f=hjk,h^g=hj,i^a=ik,i^d=h,i^f=ik,i^g=ijk,j^d=jk,k^d=j

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72340508, #72341172
maximal normal subgroup of	C₂×#72340508, #72341172
maximal quotient of	C₂×#72340508, #72340606, #72340724, …

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