#72233869

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72233780 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#85645
commutator, cosocle	C₂³×H(5,𝔽₂)	C₂²×C₃
Frattini, socle	C₂⁴	C₂²×C₂²⋊A₄
Fitting	#72233780	C₃
hypercenter	C₂³	#85645
hypocenter	C₂²×H(5,𝔽₂)	C₃×D₄
LCS	G ▷ C₂³×H(5,𝔽₂) ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ C₂³

87764 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 87764 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 96

Polycyclic presentation:

2^2,3,2^8|g^2=k,h^2=k,b^a=bd,e^c=f,f^c=ef,g^b=gj,g^c=h,g^f=gk,h^b=hi,h^c=ghk,h^e=hk,h^g=hk,i^c=ij,j^c=i

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252245, #72267141, #72267166
maximal normal subgroup of	#72252245, #72267141, #72267166
maximal quotient of	#72234205

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