#72241426

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(5,𝔽₄) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	C₂²⋊(C₂²⋊(C₂²⋊A₄))
commutator, Fitting, cosocle, hypocenter	H(5,𝔽₄)	C₃
LCS	G ▷ H(5,𝔽₄)
UCS	C₁ ◁ C₂²

144786 conjugacy classes of subgroups; only the 534 normal subgroups are shown.

normal subgroups only

This group has 144786 conjugacy classes of subgroups — too many to draw. Only its 534 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|b^2=k,c^2=k,d^2=j,e^2=j,b^a=bc,c^a=b,c^b=ck,d^a=de,d^b=dj,d^c=djk,e^a=d,e^b=ek,e^c=ej,e^d=ej,f^a=g,f^b=fk,f^c=fjk,g^a=fg,g^b=gjk,g^c=gj,h^a=i,h^d=hk,h^e=hjk,i^a=hi,i^d=ijk,i^e=ij

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72242397, #72249748
maximal subgroup of	#72241454, #72242397, #72243519, …
maximal normal subgroup of	#72242397, #72249748

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